INVESTIGATION OF MODES OF THERMAL FRACTURE

OF SOME BRITTLE MATERIALS

by

CHARLES R. NELSON

BS, University of North Dakota(1962)

.MS, Massachusetts Institute of Technology(1965)

CE, Massachusetts Institute of Technology(1967)

Submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

at the

Massachusetts Institute of Technology

September 1969

Signatu"re of AuthorDepartment of Civil Engineering

Certified by

Accepted by

................... ..- .Thesis Supervisor

Chairman,Departmental Committee on Graduate Students

\

ABSTRACT

I~VESTI~ATION OF MODES OF THERMAL FRACTURE

OF SOME BRITTLE MATERIALS

by

CHARLES R. NELSON

Submitted to the De~artment of Civil Engineering on Septem­ber 2, 1969, in partial fulfillment of the requirements forthe degree of Doctor of Philosophy.

An analysis of the temperatures and the thermalstresses in a detailed and accurate manner with the finiteelement technique resulted in the following advancementsin the understanding of the nature- of the thermal spallingproblem.

Thermal spalling due to local heating on the surface·of briLtle materials is the result of the comple~e prop­agation to total failure of a fracture induced by tensilestresses acting on a plane parallel to the surface and justunder the heated area. If the material at th15 surfacefails in compression before the tensile stress inducedfracture occurs, spal11ng is prevented. There must besufficient strain energy released by the fracture processto form the fracture surface required for complete failule.

Experiments conducted on granite, marble and porcelaincorroborate the theoretical expectations. The granite andporcelain specimens spalled at the heating time whichproduce calculated tensile stresses equal to the tensilestrength. Crushing of the surface was observed on themarble specimens as predicted from the calculated stresses.

Convex surfaces and a high ratio·of compressivestrength to tensile strength increase the likelihood thatspalling will occur in a given situation.

Thesis Supervisor: .Fred Moavenzadeh

Title: Associate Professor of Civil Engineering

2

ACKNOWLEDGMENTS

The initial part of this thesis was carried out

under partial support by the National Science Foundation

via a fellowship to the author. Later the U. S. Department

of Transport supplied the complete support. This suppo~t

1s gratefully acknowledged.

Appreciation is extended to the thesis committee

members, Professor Pian, Professor Con~or and Professor

McGarry for their interest and advice. Special appreciation

Is extended to my advisor, Professor Moavenzadeh, for his

advice, assistance, patience and encouragement over an

extended period of time.

Fellow student, George Farra provided his valuable.

aid and abilities which contributed greatly to the

suc~essful conception and execution of this thesis.

I am indebted to my children, Brent and Kimberlee,

for the opportunities and experiences we have missed and

lost forever while I was working on this thesis. Deserving

credit for perseverance is my wife, Marlys, who in addition

to cheerfully typing this manuscript, worked part time,

made financial sacrifices and did without my company.

3

TABLE OF CONTENTS

TIT LE PAGE •••••••.•••••••••••••••••••••.•.••••••••• 1

ABSTRACT •••••••••• '. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2

AC KNOWLEDGMENT ••••••••••••••••••••••••••••••••••••• 3

TABLE OF CONlfENfJ.1S •••••••••••••••••••••••••••••••••• 4

INTRODUCTIONCHAPTER ,I

1.1 General .......................... 7

1.1.1 Basic Mechanisms 8

1. 2 Obj ect1ve and Scope ••.•.••..•••.. 11

CHAPTER II THEORY

2.1 General .......................... l~

2.2. Fracture Mechanics ......••... ~ ... 14

2.2.1 Past Work on RockDestruction .......•..•...• 15

2.2.2 Related Past Work , 11

2.2.3 The Problem of.Spalllng 22

2.2.~ Initiation and Propagationof Fracture ...........•... 28

2.2.5 Failure Mechanisms forSpa11ing 39

2.3 Analysis Technique 48:t

2.3.1 General................... 48

2.3.2 Characteristics of theMaterial Properties 51

4

TABLE OF CONTENTS(Continued)

Page

2,3.3 Unusual BoundaryConditions 52

2.3.4 Sequence of Operations 53

2 . 3· 5 Heat ing Time 53

2.3.6 Numerical Methods Used 54

2.3.1 Heat Transfer Analysis Sij

2. 3. 8 Stress Analysis .•......•.. 56

2.3.9 Energy CalcUlations 57

CHAPTER III EXAMPLES

. 3.1 Purpose .......................... 58

3.2 Material Properties •..•.......... 59

3.3 Experimental Evaluation of someProperties 62

3.4 Specific Examples with Thin Disks. 63

. 3.4.1 General •....•••..•..•••... 63

3.5 Specific Examples with Cylinders. 69

3. 5 .1 Generai................... 69

3.5.2 Variation in .SpecimenLength .•••.••••••"• • • • . • • •• 72

3.5.3

3.5.4

3.5.5

•Variation in Power(Granite) .....•...........

Variation in Power(r·1ar b1e) .

Variation in Power(Porcelain) .

5

76

79

87

Page

D. List of Figures ••••..••••.•••••.. 149

101

102

152...................

......................

......................

6

Conclusions

Future Work

List. of Tables

TABLE OF CONTENTS(Continued)

A. Heat Transfer by the Finite ElementMethod ...•.•..•...••.•.•.••.•••.. 111

5. Strain Energy Calculations , 126

C. Detailed Instruction for theAnalysis Technique .•••.•••.•••••. 126

E.

CHAPTER· IV CONCLUSIONS AND FUTURE WORK

APPENDICES

BIOGRAPHY .......•.............................•••.. 109

BIBLIOGRAPHY AND REFERENCES ..............•...•••••. 107

I

-

CHAPTER I

INTRODUCTION

1.1 GENERAL

,Today an increasing amount of :~ock 1s being excavated

for construction and mining projects. New methods are

bein~ introduced to increase the excavation rates, reduce

the cost or both. Some of these methods are modifications

of older methods while others are essentially new or

novel. The design and use of the equipment for these new

methods 1s at the present difficult because of lack of

knowledge of what the requirements are'.

An exampie of a modification of an older method of

excavation is the boring or tunneling machines for large

tunnels which use rolling cutter wheels in much the same

manner as do the smaller drills that have been used for

sometime. The design and operation of these large machines

can be based, in part, on the large amount of data and

experience which has been slowly. and expensively accu­

mulated from the operation of the smaller drills. However,

for many of the new methods being developed the empirical

data and experien~e are totally inadequate for proper

designing of equipment. Typical of these methods are the

use of shaped explosive ~harges, electrical heating elements

7

and combinations of older methods such as simultaneously

heating ~nd us~ng convential cutting rollers.

There lsa need to develop a rational technique for

the design and use of the excavating equipment for these

new' methods~ since the alternative to the development of

such a lational technique 1s the establishment of large

amounts of empirical data with extensive and comprehensive

full scale tests. This would be costly and time con-

Burning, whereas a rational technique should only require

a few full scale experiments to establish its validity

and range of applicability. The basis of such rational

techniques for the design and use of new equipment would.

be a quantitative understanding of the mechanisms within

the rock which result in the removal cf the material.

This would allow the engineer to design equipment to most

effectively activate those mechanisms. For example, frac­

ture 1s one class of mechanisms which are Important in the

removal of rock by spalling due to rapid heating of the

surface. A detailed and accurate knowledge of the stress

field that causes this fracture would be required before

the equipment which generates the stress field could be

rationally and correctly designed.

1.1.1 B~sic Mechanisms. Most methods of excavation,

old and new, ~re based on one or more of the following

8

•

basic- mechanisms which degrade the rock to a state

that all~ws ea~y removal [1].

1. Phase Change

2. Chemical Reaction

3. Fracture

The most common method based on a phase change 1s

to employ heat to melt or vaporize the material which

will then flow away. For example, large scale use of

this 1s being made to mine sulfur. The sulfur 1s then

pumped to the surface and allowed to solidify before

further processing ..

The chemical reactions may be caused by the addi-

t10n of chemicals which will react with the rock and

result in products which can be simply and easily removed.

Also in this category are the chemical reactions which

occur due to heating the material., An example of this

1s the decomposition of calcium carbonate in marble which

results in a soft crumbly product that is easily removed.

Methods which-are based on fracture have the most

wide spread use today and are usually the most efficient

in terms of cost and energy absorbed per unit of rock

removed. This energy is dissipated within the rock during

9

fracture by plastic deformations, formation of new 5ur-

faces, h~at1ng~ shock waves, kinetic energy of chips and

others. For mechanical drilling methods it has been estab-

11shed that the energy required is approximately inversely

proportional to the final size of the removed rock [ 1 J.

Therefore, the methods which produce the largest chips

requires. the least energy input to the rock. However,

since different methods have different energy conversion

efficiencies and may use different sources of ~nergy,

the one which uses the least energy 1s not necessarily

the least expensive.

The fractures in the rock are induced by the stresses.

which exist in the material at the start of fracture.

These stresses are usually induced mechanically by apply-

lng concentrated loads at the surface, or thermally by

heating or cooling the surface. Also, of great importance

are the stresses which exist in the rock due to natural

causes which may make the fracturing process easier or

more difficult depending on thei~ pattern and magnitude.

For example, in very deep wells the high existing stresses

make the rock behave in a plastic manner which makes

fracturing more difficult. The size and shape of rock

pieces removed depends on the type of rock and the method

used.

10

The use of methods based on fracturing the rock

by thermal stresses 1s gaining wide spread acceptance

today in rocks which are exceptionally hard to drill

by convent1al methods [1]. A prime example is the

use of flame jets to drill the blast holes during the

mining of the iron ore, called taconite. The flame

jets are. also being used to channel blocks of granite,

thus eliminating blasting operations which tended to

degrade the material that 1s used for building and

monument use.

The actual mechanisms by which the flame jet removes

the rock during spalling is not understood in any detail.

[2 ], although it is usually assumed that it 1s a thermal

stress induced fracture. The rock removal rate decreases

and in some cases nearly stops in some rocks and in the

region of faults and other discontinuities. The addition

of an abrasive to the flow of hot gas increases the

removal rate in some soft rocks [2]. It is not well

understood why hard rocks usually spall best and some

rocks do not seem to spall at all.

1.2 OBJECTIVE AND SCOPE

The objective of the thesis is to develop and dem-

onstrate a practical method to study in a quantitative

manner the basic failure mechanisms of- the brittle fracture

11

type· in rock, during excavation. This will help in the

rational design of excavation equipment by eval~ating the

effect of the applied loads or heating in an analytical

manner.

The scope of the thesis is:

1. The development of' an analysis technique to eval-

uate the" temperatures and stresses induced in the rock with

sufficient detail and accuracy that the mechanisms of fail-

ure of the brittle fracture type may be investigated. It

1s capable of handling complicated boundary conditions and

nonlinear and nonisotropic material properties.

2. The comparison of results fI·om the analysis with•

experimental data to validate the analysis technique and

the assumptions which went into it.

3. The development of hypotheses about some of the

important fracture mechanisms of failure which may occur

when rock or other brittle materials are heated rapidly

on the surface. These hypotheses are sufficient, for

many cases, to predict if spalling results for a given

set of conditions. These hypotheses are based on the

material's properties and the temperatures and stresses

evaluated with the developed analysis t~chnlque.

~. The experimentation on spe~imens of marble,

granite and porcelain using a gas laser for a heat source.

12

The developed hypotheses are applied and comparisons

made between predicted and experimental resul~s.

The finite element method was used for the heat

transfer and stress analyses so that it was possible

to solve problems having irregular boundaries and

nonlinear and nonisotropic material properties. Th~

heat transfer analysis computer program evaluates

the transient temperatures throughout the body based

on the initial conditions, boundary conditions and

material properties. The stress analysis computer

program evaluates the stresses based on the tem­

perature distribution, boundary conditions and material

properties. Both computer programs are limited to two­

dimensions in order to minimize time.

The theory of spalling is based on the concepts or·

modern fracture mechanics: It expresses the spallability

of a material, i.e. criteria for crack initiation and

crack propagation) in terms of the material properties

and the functions evaluated with the analysis technique.

Experiments were conducted on specimens of marble,

granite and porcelain in the shapes of disks·, cylinders

and spheres. The. specimen was locally heated using a

carbon dio~ide gas laser.· The experimental data and

results were compared· with results predicted by the

analysis and theory to check and verify both.

13

I

CHAPTER II

THEORY

2.1 GENERAL

. The theoretical development in this thesis 1s dlvided

into two parts. The first is the development of hypotheses

abo~t the fracture mechanisms for thermal spalling, The

second 1s the development of analysis techniques to cal-

culate the temperatures and stresses for use in the first

part.

2.2 FRACTURE MECHANICS

To date,.most of the research into the destruction

of rock for mining or construction purposes has not

investigated the mechanisms of failure in detail. It has

instead been experimental parameter studies using full

scale; or nearly full scale equipment, or "theoretical"

studies based on assumed stress fields or overly sim-

p11fied models [3]..

The results of this work has been very valuable in

reach~ng local optimums in the design .and the operational

technique for the existing equipment. However, since

detailed accurate knowledge ~f the mechanism or mechanisms

of destruction was not available, it was difficult to apply

l~

the results o~ this type of research to other materials

or excavating equipment which were different in even

modest ways.

The mechanisms were not investigated in detail because

the stresses induced within.the material by the equipment

was not known with sufficient detail or accuracy nor were

the basic principles of rock failure well understood.

There were at least two reasons why the stresses could not

be found with the required detail and accuracy. First, the

boundary conditions that existed between the rock and the

excavating equipment were not known with the detail and

accuracy required for analysis. Second, due to the complex

boundary conditions and the nonlinear material properties,

an analysis was impossible with the classical methods.

2.2.1 Past Work on Rock Destruction. Evens and

Murrell [ 4] modeled the problem of a wedge penetrating

soft coal in a simple manner. They assumed that the

normal stress between the wedge and the coal was equal to

the unconfined compressive strength of the coal and that

the tangential stress between the wedge and the coal acted

. upward on the surface of the wedge. and was equal to the

normal stress times the coefficient of friction. This

simple model resulted in predictions which correlated well

15

with experiments conducted in soft coal but very poorly

with experiments conducted in limestone, a more brittle

rock in which chipping 1s ~mportant.

Paul and S1karskie [5] modeled the problem of a

wed~e penetrating a brittle material in which chipping

would take place. Their assumptions were that the Mohr­

Coulomb criteria was satisfied along the fracture plane

at the time of chip formation, and the fracture ext~nds

1n a straight line from the tip of the wedge to the

surface of the material some uistance away from the

wedge. This model accurately predicted the shape of

the loading curve and the required penetration force

if the slope of the load-deformation curve was found

experimentally.

Several studies using the technique of photo-elastic

analysis to find the stresses have been conducted. Ones by

Garner [6] found that there existed tensile stresses

across the potential fracture plane for the case of in­

dexed penetrations by a single bit-tooth. However, since

the materials used for the photo-elastic analysis have

different properties than real rock and the interaction

with the bit is hard to simulate, littl~quantitative

work has been done.

16

2.2.2 Related Past Work. Research conducted by non­

mining ortentented groups has investigated the destruction

mechanisms 1n some detail. One reason they have been able

to do this is that the loading, material properties and

geometry were often more favorable to a simple, but accurate

and detailed analy~~s. An example of the experiments which

they conducted was to submerge spheres of homogeneous

material into hot fluids and observe the resulting fracture

phenomena.

In particular, people in the ceramics industry have

been interested in the phenomenon called thermal spalllng

because of the many products that must be designed to

withstand rapid heating or cooling. They have been in­

terested in finding the actual mechanisms of failure so

that products of ceramic materials could be rationally

designed to withstand conditions for which it was not

practical, or economical to run experiments on pro-

totypes. At first their efforts were not successful

because a good analysis was not possible due to the complex

specimen shape used to study the problem. This lead to

erroneous conclusions as to the mechanisms responsible for

the spalling. An example of this was the work by Norton [7]

in which he incorrectly hypothesized the cause of spalling

to be a fracture induced by shear stresses which he thought

17

existed along the isothermal lines. He even derived

equations for these imaginary shear stresses, b~t they

were not valid. His experiments consisted of rapidly

heating one end of oven liner bricks and noticing loss

of material at the heated end due to spalling. (See

Figure 1). Initially for analys1~ Norton modeled the. .

problem with a half-space which was .subjected to an in-

crease in surface temperature. It was the lack of tensile

stresses for this model that lead him to misinterpret the

mechanism to be a shear stress induced fracture. Later

[ 8] he conducted a photo-elastic study of heating of

brick shaped models. However, he again erroneously•

interpreted the results to justify his shear stress

mechanism.

Norton [ 9] continued to put forth his shear stress

1ndu~ed mechanism theory in one form or another. Norton's

theory was rebutted by Preston [10], who suggested that

thermal spalling is always caused by tensile stresses wlth-

in the material. However, he was ~ot certain of the

location of the tensile stresses that started the' spalling

but indicated that the stress may be such that the crack

would start at the surface and travel into the body.

Preston also stated that the sJ~rain energy released

by the propagating crack must exceed the energy required

18

·... I •

j""'~T'<':,;~- ~".- <?~: '. :;~.~.. ~\...."'..,. ~., .. "'. '.".'"'- - " ~-; _. ,._.. ."."': ,- .-- '0. ---'- .- ~ I

t

~

\.0

.~~E SURFACE~

I .Lx

THIS END HEATED STRESSES NORMALTO THE CENTER

FIGURE I: SPALL SHAPE AND ASSOCIATED STRESSES [7]

to form the new surface area. In this respect, he se,ems

to have been the first to apply both the concept of crack

initiation and, crack propagation to the spalling problem.

Preston did not conduct any analysis or experiments to

back up this theory about the mechanisms of spal11ng.

In 1955 Kingery [11] concluded, based on the inves­

tigations to that date, that the mechanism responsible

for thermal fracture was a stress induced fracture. Based

on this idea, it was possible to establish the,effect that

the values of the material properties have upon the spall­

ability of a material.

Since a stress initiated fracture was the mechanism

which causes spalling, it was shown that materials with

high strength and a low modulus of elasticity would be the

most spall resistant. Also, for the case of spalling

caused by sudden surface heating, materials with a high

thermal conductivity and dlffusivity, and a low coefficient

of thermal expansion and emissivity were the most spall

resistant.

In 1963 Hasselman [12J again brought up the hypothesis

(first pointed out by Preston [10]) that the strain energy

released by the crack's formation must exceed the energy

required to create the new surface area or else the crack

will be halted and spal'ling will not occur.

20

Hasselman conducted experiment,s with specimens, of a

very brittle material, a less brittle material and a tough

material. These materials were rapidly heated to observe

their spal1~ng behavior. The brittle material shattered

when heated, the secon~ and third ~ld not break completely

apart. However, they may have" had spall fractures within

since it was calculated that the tensile stress of each

material was exceeded in the interior region of the

spheres. His calculations show that for the two tougher

materials there was not sufficient energy available for

release,during fracture, in order to create the new surface

area required to completely separate a' "chip" from the

parent material.

The conclusion to be drawn from this work is that there

exist two conditions that are necessary and sufficient to

incur a spalling type of failure: 1) The fracture ~trength

of the material must be exceeded at some point in the

specimen. 2) The releasable strain energy associated with

the projected crack formation must ~xceed the energy

required to form the new surface area created by this crack.

To minimize this available energy for a spall resist-

ant material, the. strength should be low and the modulus

of elasticity and effective surface energy should be high.

21

The effective surface energy rere~red to is the energy

required to create a unit area of new surface by the

fracture process:

Before this work by Hasselman, other efforts to find

what a material's properties should be to prevent spalling

had considered only the initiation of the cra~k. Their

observations had never included a situation in which a

crack was observed to form, but did not 'propagate a

sufficient distance to form a spall. The reasons for this

oversight were as follows: 1) The crack usually starts

in the interior of the specimen and is difficult to find

if it has not propagated to the surface. 2) Most

materials which the other people tested have properties

such that if a crack starts there is sufficient energy

available to completely propagate the crack and thus

cause spalling.

2.2.3 The Problem of Spall1ng. The phenomenon,

commonly known as spalling, occurs in many different

materials and situations. Often it· 1s desirable to control

the phenomenon for the advantage of man. Some of these

situations will be considered to gain insight into the

problem and to demonstrate that there is ~ marked

similarity between the stress fields for the various cases.

22

· First, th~ term spalling will be defined as the loss

of material from a body by a fracturing process which is

induced by the stresses existing in the. body. These

stresses may be due to mechanical loadings, temperature

changes, phase changes, changes in moisture content, etc.

While 1n most'cases the final stresses are the result of

some combination of these, 'often on~y one will predominate.

The spal11ng of fire bricks has long been a problem

in large industrial ovens. It was this problem that lead

Norton [ 7] to conduct his research in the field of

spalllng. The end of the brick which formed tile inside

surface of the oven was the location at which the spalling

occurred. This end would be exposed to rapid heating

when the oven was started inducing stresses as shown in

Figure 1.

Spalling in concrete 1s a problem in sidewalks, roads,

buildings and most other structures made from that material.

There are two locations at which the spalllng usually occurs.

The first 1s at joints which are not separated an adequate

distance or do not have an appropriate soft filler in place:

When' a joint with either of these shortcomings tends to

close due to loading, tempera.ture change 6r others, large

stresses will be locally transmftte~ across the joint as

shown in Figure 2 causing a spall to form at the edges.

23

The second place that spalling commonly takes place in

concrete is at· the surface, around a piece of porous

aggregate: If the aggregate 1s filled with water whpn

subjected to freezing temperatures, stresses are

developed as shown in Figure 2 causing the spalling of

material around the porous aggregate.

When some types of rocks and other materials are

rapidly heated at the surface, spalling occurs. This

has been observed In detail by the author while heating

materials with a gas laser. The associated temperature

anJ stress distributions are shown in Figure. 3.

Spalling'is also observed to occur in some materials

during compressive teSts. This spalling usually occurs

at a compressive stress which is approaching the compressive

strength of the materials.· A hard inhomogeneity in the

material being tested would cause a local stress dis­

tribution as shown in Figur~ ~.

In each of the cases cited above there 1s great

similarity among the various stress fields associated with

each of the spalling regions. In all cases, there extst

high compressive stresses near the surface and parallel~

to it. Also, in each case there were tensile stresses

acting on a plane parallel to and at a small distance

below the surface.

24

"J - ••J ILl 1 1111_AIIJftfl_~L.~;;;~~:~···~ ~,

:.<~

~- 0 .-_. .."", l' . ~

EXPANDING .AGGREGATE

: 0 ,,~. 0, ~'. 'e:0 -, : -b- · , 4• .' - I· '.. 0 -' • ,-

'. .' A'.· · -. •. 0 '. -0•• "_ V • 0 '. • • ,

, 'Ii '. '. : o. · · ,0 · 'D 'p', . , ~, ' 'po' - · -.. , :• .''''.' 0 • OJ I • ", ;

• I • a.··. 'I ~t:'·.6·

"I): · · '0'0· 0

, • • .. • ~,................ W'

POTENTIAL·

• SPALLINGCONDITIONS ~

f'..>\Jl '.~' . ' ,. _() ';, . Q • . - f"~

, '0- · ". • ", .-'I I - , 0 I • .,, · c.'o ~ -(). e I r.

I •

, I 0 "' · · ·I • •••• ~ ••••. .. .o"e ••

- : -=- · . '.0

aACIING

STRESSES

..• C>"•.. 0 .. 0 I··t~. · · .' '. (J

'_0.'0 '(J'• .' .' : I .' .- •• D'. -. '

· .' O· 0 ".t:i eo • •

• • •• • tI/I • •. " . ,;

'0 · '"· '.'0 'a'. .. .

JOINT IN CONCRETE

II

EXPANDING AGGREGATE

FIGURE 2= POTENTIAL SPALLING CONDITIONS IN CONCRETE

. II . II OJ JI .1."f"~1i)!tv:t~:'~;·. ,:, ',' !''-I'*1

-~;

_ .. 1 •.'--'

•

(COMPRESSION)

~4- LASE~ RADIATION

T1\.) 1-5"

0'\ i l"-../9

I -----~< ~~1'511~

ENLARGEDSECTION <l

FIGURE :3: THERMAL STRESS DISTRIBUTIO\I FROM LASER RADIATION

27

-

bm~enfflH:~8

.e

~;.~

I~-z:::>-'<tZ.-~0Z••

zV

0W

b§

b0::

0:::>

~C

)-

0LL

a::~

The material for each case was brittle in nature

from which little or no yielding could be expected.

2.2.~ Initiation and Propagation of Fractures. The

study of the failure modes or brittle materials with

consideration given to initial flaws was first considered

by Griffith [13]. He was considering the problem of the

tensile strength of glass being much lower than theoret-

ically expected. He postulated that there were initially

very small natural cracks existing in the glass which

produced stress concentrations at their tips, thus

causing the critical crack to propagate. This resulted

in total failure at a nominal tensile ·stress, much lowerI

than theoretically expected.

When the crack propagated, ene~gy was absorbed by

the process which formed the new fracture surfaces.

Griffith attributed this surface energy to the rupturing

of the bonds between the atoms on each ,side of the

fracture. For most materials, other than the most brittle,

such as glass, it has since been fo~nd [14J .that a

significant amount of the apparent effective surface en~rgy

is absorbed in plastic flow and formation of branch cracks.

In addition, if t~e fracture propagates at high speed,

energy is lost in the production of sonic waves [14].

28

-

The energy required to create the fracture surface is

found exp~rime~tally in simple experiments in which the

propagation-rate 'can be controlled and the total energy

input measured. This total energy is divided by the total

new' surface area formed to arrive at a figure called the

effective specific surface energy, l~

The source of the energy for the propagation is

conveniently divided into the strain energy released by

fracture and the energy added during propagation from

sources outside the body. Griffith initially considered

only the released strain energy.

Based on the assumption that the crack was penny-

shaped, Griffith derived the following equation of the

nominal uniaxial strength of a brittle material 1n terms

of the crack width and the effective surface energy_ It

neglects outside sources of energy.

2

a = n y E / 2 c (l-~ )f

Of The nominal uniaxial tensile strength normal to the

crack.

c The radius of the penny shaped crack.

E,~ - Modulus and Passion's ratio.

y Effect~ve surface' enery per up.i t of area.

29

30

propagate.

if ~1o=01 + a. f

For the simple case of uniaxial tension normal to

2

(Oi- o~) - BOf (01+ 02) = 0 if 01+ 302 > 0

For fractures caused by more complex two-dimensional

been derived [3]. For shear fracture it is:

with the crack was added to the energy consumed by crack

growth, and the sum was differentiated with respect to

Extension·rracture:

the following manner: First, the strain energy associated

crack growth equals that absorbed in formation of new

this is saying that when the strain energy released by

the crack length C, and set equal to zero. Physically,

°1 and O 2 are the principal stresses (compression Positive)

and 0 is the uniaxial tensile strength. They are shownf

graphically in Figure 5.

fracture surfaces, the crack will be unstable and

the crack, the resulting fracture is referred to as an

stress disturbances, the following .failure criterion has

surfaces parallel to each other during propagation.

This equation was derived using energy considerations in

extension fracture. Another, called shear fracture,

.occurs when there exist stresses which slide the two

n U•••BLlliJdiaa_

.. , ...._•• ~ _~ __ • . ..., • ~=-:.i '.-~ ~:--':";-"':-l-~.a.-....;I... ~;..., ~l'~."·:-: ....~.!!t~":"~f~~~·~~~~~:*~~:tiJ;..~~~")',~!,:-·",,-~1P'~~\~;l ...:~ ..C.Jr,:~~....~ ~ ._'~:.I t- ~~•.• ~ ~

0'2

OJ ,. a2 - PRINCIPLE STRESSES

T - UNIAXIAL TENSILE STRENGTH-

Wf-J

RANGE OF I RANGE OF•• IF EXTENSION ... SHEAR ----

FAILURES FAILURES

FRICTION ACROSS CLOSEDCRACKS CONSIDERED

T

T

3T

-,-...---Oi

FIGURE 5: BRITTLE FRACTURE CRITERION IN TWO DIMENSIONS [ 3 ]

32

W-JQ..

~U52- r--,

to

.-..........

••CDW0::::)t')[i:

33

r

This is the same as the condition for the two-dimensional

case.

°3 + a = af

°3 = minimum principal stress (max-

imum tension)

or = uniaxial tensile strength

To date, consideration of the fracture process has

been confined to the case of homogeneous stress fields.

Since most problems encountered in reality have non­

homogeneous stress fields, they must be considered in the

study of fracture failures. The influence of the

nonhomogeneous stress on the initiation of the fracture

would be negligible for many cases because the initial

flaws are usually very small and the stresses would

change very slowly with respect to the flaw length. Thus,

the initiation' would occur under conditions similar to

the case of a homogeneous stress distribution. However,

the fracture may propagate over a aistance in which the

stresses vary in a substantial manner. The direction

and distance the fracture travels must be known in order

to accurately predict such types of complete failure as

spalling.

The direction that the fracture takes will be dependent

34

on the instar.taneous stresses at the moving tip. The~e

will be ~ function of the initial stresses,. additional

loading since crack initiation and the changes in stresses

brought about by the fracture. The fracture will

propagate in the direction of the line on which the maximum

tensile stress acts at the tip. For many cases, the

knowledge of the prefracture (initial) stresses and the

initial fracture path will allow a very good qualitative

estimate of the direction of further propagation. If

this is not possi~leJ a detailed analysis must be used

to evaluate the new stress distribution with due regard

given the new boundary conditions. Experimental observa-.tions of the exact path of fracture should be done

Wherever possible.

Once the expected direction of the fracture is known,

it must be established if there is sufficient energy to

drive the crack the fUll distance causing the failure

anticipated. This evaluation must be quantitative in

nature and ideally would be made i~ the following manner:

The rate that at which energy is made available as a

function of fracture propagation would be calculated and

compared with that energy required for c~eation of the

new r~acture area. ·If the available energy exceeded that

required, the propag~tion would continue. The energy

35

avai~able for further propagation would be that released

by such propagation, and any energy left over from earlier

propagation which was not used in creating new fracture

area or lost by wave motion to distant parts of the body.

This excess energy would be stored locally in the form

of kinetic energy and would be lost if the speed of

propagation was not continuous and rapid.

To state the problem more formally, consider G as

the energy made available by a unit area increase in

fracture surface. Much of this energy G is made

available by the release of strain energy from the

reduction in stresses due to the fracture. if it is

an extension type fracture which penetrates into a

zone where compressive stresses are acting across the

fracture then G will be negative. Following this,

consider additional notation for other relevant

parameters: y is the energy consumed to ·create a

unit of fracture area; TG is the total energy made

available if the fracture propagates to completion; Ty

is the total energy required to form the new fracture

surface; LG is energy lost during the creation of a

unit of fracture area and TLG is the total lost during

the creation of the whole fractu~e. A is the totalo

new fracture area formed during propagation to complete

36

is A is:

y dA

LO dA

G da

G dA - JY dA - J LG dA

A A

Ao

I

=

=

TLG =

Ty

TG

NG =

NG = f (G - Y - LG) dA

A

If NG is positive for all values of A. it can be

failure. The relations between these quanities are:

The NO net energy at a given time when the fracture area

expected that the fracture will propagate. to completion.. "

If NG > 0 for all A from 0 to A completeo

propagation occurs.

31

fracture area is:

38

The second relation which can be established is the

at initiation

Y - LG > 0 for any point

G - y - LO ~ a

tf NG = 0 and G

The evaluation of these energy relations for the entire

The prev~ous relation implies an excess of energy

The total energy made available is found by evaluating

Ty = J Y dA

Ao

points. Expressed mathematically, this is to say that

would continue if the available energy was zero at discrete

available at all times. It is also true that propagation

and NG > 0 for all other point~

then complete crack propagation will occur.

the new area:

the scope of present analysis techniques. For most cases

fracture area for most practical problems would be beyond

there would only be two relations known in a quantitative

manner. The first is at initiation where the rate of

available energy equals or exceeds that required to form

total energy absorbed in fracture work and the total made

available. The total energy absorbed in creating new

-

~ ,

39

TG < Ty + TLG complete fracture is impossible

For most2.2.5 Failure Mechanisms for Spalling.

propagation to complete failure.

only two quantitative relations can be established. These

are the last two equations in Section 2.2.4. One is the

TG > Ty + TLG necessary' (not sufficient) to

A necessary, but not sufficient, condition for complete

and TLG 1s the total energy lost o~tside the system during

TO = PE initial - PE final + IE

practical problems with nonhomogeneous st~ess distributions,

where TG 1s the total energy made a~ailableJ Ty 1s the

total energy used in creating new fracture surface area

complete fracture

the new fracture surface for complete failure.

exceed that which is lost plus that which 1s used to form

terms. The total energy made available must equal or

fracture may then be established between the final energy

during the fracture propagation process.

where PE 1s the potential energy and IR is the energy input

potentia~ ene~gy and then adding any energy input to the

system during prbpagation:

the initial potential energy, subtracting the final

-

the uniaxial tensile strength of that material. If the

fracture is initiated in a plane approximately parallel

There are several possible types of fracture which

This stress distributionthe manner shown in Figure 1.

bottom of the chip. (See Figure 8 )

dirpction, with a slight preference to approach the surface

on account of "the bending stresses ·being tensile on the

will cause further propagation in the same general

The energy requi~ed to drive tre fracture would come

to the surface, the stress distribution will change in

acting in the zone behind the hot spot will initiate an

extension type of fracture in the material if they excee~

Considering the thermal stresses in Figure 3 , which were

due to local heating with a laser, the tensile stresses

completed fracture must run under and up to the surface.

requirements of spalling are that the fracture completely

sepa~ate materials from the ~ain body. To do this the

may initiate potential spalling situations. Of these,

only one leads directly to the formation of a spall. The

itative information about fracture direction, must be used

in hypotheses about overall failure rne.chanlsms.

ener~y·balance. at fracture initiation and the other 1s the

overall energy balance. These, together with the qual-

'I:·,f"~ r •

;

I

~~~~~1:,:;ij"·-it

~ RADIANT HEAT

.....

I

ENLARGEDSECTION

r::::::::a"-... FRACTURE

~

~

t. ,

FIGURE 7: STRESSES AFTER FRACTURE INITIATION

DURING THERMAL SPALLING

'M £ ? U If _

. . ... ."

~

ru

.THE STRESS DISTRIBUTION SHOWN IN

THIS FIGURE IS NOMINAL.

THE ACTUAL TENSION AT THE TIP

OF THE FRACTURE WILL BE MUCH

GREATER THAN SHOWN DUE ro THECONCENTRATION FAClOR.

~p OF FRJCTURE ~Ji+-TENSION•

I

t"

FIGURE ·8: NOMINAL, STRESS DISTRIBUTION

NEAR TIP'OF FRACTURE

from the release of stored elastic strain energy and

further heating during the fracture process. If the

releasable elastic strain energy 1s initially sufficient

for complete formation of the spall, the propagation

will be at such speed that the energy contributed by

further heating would be negligible.*

If the fracture had initiated in a v~rtlcal direction,

propagation would cease upon entering the compressive zone

above or the large very slightly 'stressed zone below. (See

Figure 9 )', Such vertical fracture does not significantly

reduce the tensile stresses acting on the planes parallel

to the surface.

Fracture may also be initiated at the surface where

the compressive stresses are the greatest. (See FigurelO).

A failure of this type would tend to reduce the tensile"

stresses underlying the compressed zone. It 1s thus.

possible and probable that there would never be a tensile

failure and hence spalling would never occur .

. For the thermal stress problem, a compressive failure

at the surface would increase the porosity of the surface

* The released strain energy, greatly in;excess of thatrequired, is converted into kineti~ energy which resultsin faster propagation.

~3

+-TENSION.

.am~ RADIANT HEAT

,\ POSSIBLE

FRACTURE

/'COMPRESSION

RADIAL STRESS SI-IONN .

~

I ENLARGED

.!=:" .... ISECTION

<l

FIGURE 9: VERTICAL CRACK IN THERMAL SPALLING

..t=­Vl

.mn""'- RADIANT HEAT

--..--

'CRUSHED ZONE ~

. COMPRESSION

RADIAL srRESS SHOWN

",

I

1--~ .

¢..

ENLARGEDSECTION

FIGURE· fO: FAILURE OF SURFACE

material, which would decrease the thermal conductivity,

resulting in less heat penetrating into the intact zone.

This would reduce the existing stresses, preventing

spalling from ever occuring. Accompanying the crushing

would be possible melting at the surface by the large

amount of·heat being retained there. This melting would

result in loss of the heat energy in fusion, and in

radiation, conduction and convection to the surrounding

medium.

The circumferential tensile stresses surrounding

the hot spot could initiate a fracture which then

might propagate down into the body and" ~ideways into

slightly stressed zones next to the hot spot, and

finally into the hot spot which is in compression. Thus

this crack would not cause spalling.

The initial lack of ,energy to completely propagate

a fracture may be overcome by additional loading or

heating. There 1s at least one important potential

application of this concept of increased loading to cause

complete propagation, i.e. failure. This is in heat

assisted mechanical destruction, where the mat~rial is

heated before being subjected to mechanical loads. If the

heating injtiates the fracture, the mechanical loading may

complete the propagatiori, or if the sharp bit or the

46

mech~nical tool initiates the fracture, the heating with

its large areas of highly stressed material may supply

the energy needed to propagate the fracture.

Materials which are not completely brittle, but

undergo scme plastic deformation before fracture occurs,

will require complex analysis for the stress state at

time of fracture initiation. The evaluation' of the

available energy will also require careful consid~ration

due to the nonlinear nature of the problem.

Detailed consideration of the stress that causes

the fracture initiation and the energy for the prop­

agation will be given to specific problems o'r thermal

spalling in Chapter III.

~7

2.3 THE ANALYSIS TECHNIQUE

2.3.1 General. The analysis technique was developed

to evaluate· with- sufficient accuracy and detail the func-

t10ns necessary for predictions of the spalling behavior

of'rock and other brittle materials. It has a large degree

of flexibility and generality so that solutions can be

found for complex problems having both thermal and mechan-

1eal loads, irregular boundar~ shapes and conditions and

nonlinear material properties.

Spal11ng is caused by stress induced brittle fracture

1n the material. Under the conditions and range of

stresses which spalling takes place the material properties.

are not stress dependent. The materials in which spalling

1s a problem are ceramics, the harder rocks and concrete.

The spalling occurs at a surface which is subjected to

low (usually one atmosphere) pressure with high local

stresses due to' mechanical or thermal surface loads causing

the initial failure by tensile fracture. Up to the point

of tensile fracture, the behavior of these materials is

known to be nearly·linear and elastic [ 3].

These same materials if loaded to failure' in simple

compression or in triaxial loading can, and usually do,

have large nonlinearities in the stress-strain relation-

ships. However, when the stresses in the body are such

48

that crushing or flow takes place before a tensile fracture,

the tensile stresses are relieved and spalling does not

occur. For these reasons linear stress analysis will

suffice for the prediction of spalling behavior.

The material properties will change with the tem-

peratures which accompany thermal spalllng. The non­

linearity of the coefficient. of thermal conductivity

affects the analysis of thermal stresses more than any

of the other properties. In general, the conductivity

of oxides, such as most rocks, decreases substantially

with increases 1n temperature to the level needed to

cause thermal spalling. The material which becomes the

hottest is at-the surface where the heat is applied and

a great deerease in the conductivity at the surface

prevents the heat from penetrating into the body. This·

heat retained at the surface furtber raises the tem-

perature of this thin layer and causes additional decreases

in the conductivity. Th~s problem is further compounded

if the surface material fails in compressiorl increasing.the porosity which further decreases the conductivity at

the surface.

It 1s recogn~zed that while it 1s pos~ible to predict

the thermal spalling behavior for simple cases based on

linear behavior for most. properties, there exist a large

49

class of problems in which it is necessary to 'consider

more nonlinear "behavior. Practical problems of this

nature are combined heating and mechanical disin-

tegration of rock, slow fatigue type fracture in a heat

cycling environment and others. Since the investigation

of these and similar problems is envisioned in the future,

it was desirable to develop the analysis technique as

gener~l as possible. This required that the method have

room for growth to the more complex cases while taking

advantage of past developments which have been proven

accurate by comparisons with experimental data. To meet

these require~ents, numerical methods of analyses for

both the temperatures and stresses were used.

The method of analysis developed for this thesis is

not in itself unique (although the heat transfer program

was developed independently before it was available in

the literature) but the solution of thermal stress prob-

lerns with the complex boundary conditions has not been

confirmed with laboratory experiments as is done in this

work.

The thermo-elastic coupling is ·assumed to be of'f

lmp?r~ance only as the temperatures affect the stresses

and not the reverse. The stress levels encountered in

rocks near a 'free surface are sufficiently low for conven-

tial sources of heating and mechanical loads that the

5.0

assumption is reasonable. This may not be the case if

blasting .with convential or nuclear explosives is used

in disinteg~atln~ rock or for fracturing materials with

very intense sources of heat such as a laser in the

pulsed mode.

The inertia forces were neglected in the analysis.

Again this is reasonable 'in light of the conventional heat

sources and· for a large category of the mechanical equip­

ment used in rock disintegration. However, there is a

significant number of rock disintegration methods in

which the inertia forces play an important role.

Examples of these are blasting, percussion drilling,

some erosion methods and others. The basic numerical

technique used could be modified in the future to eval­

uate dynamic problems w~th a useful carry-over of know­

ledge generated herein on the static problem.

2.3.2 Characteristics of the Material Properties.

The characteristics of the material properties include the

usual stress related nonlinearities ~aused by yielding and

crushing and in addition, those due to temperature var­

iations. Also, many rocks of sedim~ntary origin display

a great deal of anisotropy in their materials. Some of

the changes in the properties are brought about by the

temperature .. However, those brought about by phase changes

51

are abrupt in nature and multivalued functions of the

temperature.

The developed analysis technique has the capability

to handle these nonlinear and anisotropic properties, but

only the nonlinear thermal conductivity was considered

for the examples analyzed .

. 2.3.3 Unusual Boundary Conditions. There are two

unusual boundary conditions which will have to be consid­

ered 'for practical thermal spalling conditions and the

numerical analyses developed can simply be extended to

each. One is radiation away from a boundary with a

changing temperature and the other 1s moving boundaries.

The radiation problem is of importance for evaluation

of the net heat transfer to the heated obj~ct when radiant

heat sources are used. The relative heat t~ansfer by

radiation between two bodies is not linear with respect

to temperature, but is proportional to the difference

of their surface temperatures to the fourth power. It was

not needed for the examples in this thesis where heating

was done with a gas laser becuase the radiant heat away

from the specimen was negligible compared to the laser's

input of heat.

The moving heat source or surface must be given

consideration, respectively, in evaluating the use of

52

equipment for continuous spalling where the surface will

recede 1n.dlscr~te jumps a distance equal to the thickness

of the spall", or for equipment which moves the heat sources

over the working surface.

2.3.4 Sequence of Operations. The analysis technique

consists of a series of separate calculations of the

different functions which' control the precess under consid-

eration. The appropriate sequence of the calculations to

determine the ~hermal spalling behavior 1s as follows:

1) Calculation of the temperature distribution based on

the initial conditions and the proper boundary conditions.

2) Calculation of the stress distribution based on the

proper boundary conditions and temperature distribution

calculated in the first part. 3) These calculated temp­

eratures and stresses are compared with the values required

for failure of the material to see if failure has occurred,

and if so, what type it is. 4) If part three indicates

that an extension fracture is initiated in the proper

location and headed in the proper di~ection, a calculation

is made of the strain energy available to drive the frac­

ture and a prediction is made of whether or not the frac­

ture wjll propagate to form a spall.

2.3.5 Heating Time. In thermal spalling environments

the temperature distribution is continuously changing with

53

time due to the application of heat at the surface and its

transfer .to the more remote regions of the body which

results 1n ~onti~uous variation of the magnitude and

distribution of the thermal stresses. It 1s necessary to

find the magnitude and distribution of the stresses at the

time they initiate the failure. To do so requires that the

stresses be computed for 'time intervals surficiently small

such that the first failure mode is correctly evaluated.

2.3.6 Numerical Methods. To achieve the required

flexibility anq generality, a numerical approach called

the finite element method [18] was used for both the heat

transfer analysis and the stress analysis. With this

numerical method, the body is modeled as an aggregate of

many small, but finite size regions within which assump­

tions are made "about the behavior of the function. Based

on these assumptions) equ~t1ons are formed which relate

the ~unct1on at discrete points throughout the body. The

solution of these equations together with the assumptions

within each element define the value. of the function at

all points 1n the body.

2.3.7 Heat "Transfer Analysis. For the heat transfer

analysis) the temperature 1s assumed to undergo a linear

variation with respect to the space coordinates within each

element. The. temperature of the nodes (corners) of the

54

elements are related to each other by equations based on

the conservation of energy. These equations ar~ solved

for the nodal temperatures. The element size must be

chosen small enough such that the assumed linear variation

of temperature within each element 1s a good approximation

to the true temperature distribution.

There are two methods of solution to obtain the

transient temperature distribution. One is to use modal

analysis [19] in which the solution 1s found by super-

imposing the contribution of each of the modes which

significantly contributes to a particular solution. The

other method 1s to march the solution forward by small,

time increments based on known values of the temperatures.

The second method was used because it is more flexible

with regard to nonlinear material properties and changing

boundary shape. The complete formulation or the heat

transfer analysis technique and a listing of the computer

program 1s given in Appendix A and Appendix B.

As developed, the method can solve problems which

have.the following characteristics for the transient

temperature distributions.

1) No more than two-dimensional (plane or axi­

symmetric).

55

2) The material properties may change only at the

boundary between elements.

3) The conductivity may be different in the two

directions (x, y).

4) The boundary conditions may be

A) Adiabatic

B) Given Heat Flux

C) Radiation

D) Given Temperature

E) Conduction to a Fluid of Given Temperature

5) The boundaries must be at a finite distance.

2.3.6 Stress Analysis. To evaluate the stresses,

a finite element program was used which was developed by

others [20]. This program would analyze the stresses

for problems which have the following characteristics:

1) Plane strain, plane stress or axi-symmetric~

2~ The modulus of elasticity may be a function of

the stresses all other properties are constant.

3) The material properties may· be different in

~ directions parallel to the three axis.

4) Both the temperature distribution for initial

stress free condition and the final conditions

are known.

5) The boundaries must be finite.

56

6) The condition allowed at the boundaries are:

A) Free or fixed 1n any of the directions

parallel to the axis.

B) Applied Stresses

2.3.9 Energy Calculations. As stated in Section 2.2.4

the energy which 1s available to drive the fracture 1s the

result of the release of stored strain energy and that

added by further loading or heating. For thermal spal11ng

it is observed experimentally that the fracture process

occurs at such a rapid rate that the additional heating 1s

negligible compared to the heating which precedes the

initiation of fracture~ To obtain the ·total energy made

available for 'fracture, it is necessary to calculate the

total strain energy before and after spalling and subtract

to get the strain energy released.

However, for'spalling which is caused by sudden heating

of the surface, it is possible to make approximations which

are sufficiently accurat~ for the purposes intended. This

1s done in Appendix B.

57

. , I

L

CHAPTER III

EXAMPLES

3.1 PURPOSE

Specific examples were.studied both experimentally

and analytically in order to achieve the following:

1) Demonstrate the validity a~d accuracy of the

analysis technique used to predict the thermo-elastic

response of brittle materials, and evaluate the estimates

of material properties and boundary conditions. This was

achieved by lasing thin disks in an axi-symmetric fashion,

observing the temperatures and strains, and then comparing

these with the analytical results.

2) Investigate and study the spal11ng phenomenon .

in brittle materials subjected to rapid local heating on

the surface. This was similarly achieved by axi­

symmetrically lasing cylinders of natural rock (such as

granite and marble) and spheres of man-made brittle

materials (such as porcelain).

3) Test the validity of the theory developed which

explains and predicts spalling in brittle materials. This

was done by obtaining the calculated stre~6es 'as a function

of lasing time from the analysis te~hnique, and by knowing

the ~trength characteristics of the material involved)one

58

-

coul~ rind the, time and location at which the predicted

stress condition overcomes the strength of that material.'

Several experiments would then be performed which involved

lasing three-dimensional specimens. The time and location

of failure initiation was observed and compared to the

theoretical predictions.

3.2 MATERIAL PROPERTIES

The material properties required for the analysis

technique and prediction of the spalling behavior were:

1) Thermal Conductivity

2) Specific Heat

3) Density

. '4·) Modulus of Elasticity

5) Poisson's Ratio

6) Coefficient of Thermal Expansion

7) Tensile and Compressive Strengths

8) Specific Surface Energy

The experimental evaluation of some of these properties

was beyond the scope of this work an~ estimates were made

based on pUblished data for the same or similar material.

To establish confidence in the results of analyses based

I In addition to this the energy available for thepropagation of the fracture wa~ evaluated, and takeninto consideration.

I.

59

on some estimated data, comparisons were made between

analytical and experimental results in certain exper-

iments. Since the results compared well, the analysis

program was used with confidence to solve other problems

having the same materials and heat source for ranges of

temperatures and stresses that were similar to the

control experiment.'

The following material properties were evaluated

for the granite and marble used in the experiments:

1) Modulus of Elasticity

2) Tensile and Compressive Strengths

3) Coefficient of Thermal Expans.ion

For the porcelain, only the tensile strength was

evaluated. The specific surface energy has already been

experimentally evaluated for the granite and marble [16J.

These rocks were from the same quarries as those used in

the experiments conducted herein.

All other material properties were" taken from hand-

books and other published data. Table 1 lists the value

of each property and the source of information.

I If the analytical and experimental results had notcompared well, a more detailed evaluation 6f the individ­ual properties and boundary conditions would have beenrequired.

60

MATERIAL PROPERTIES

SYMBOL DEFINITION MARBLE GRANITE PORCELAIN

Specific heat capacity 0.42 .21 .21(BTU/lb of)

K Thermal conductivity 1 12.35 X 10-5(BTU/in. sec. OF) 24ooo.+30T 31500+21.6T

p Density (lbs/in3 ) .097 ~~] .094 Ibs .094 Ibsin3 1n3

Linear coefr1cient or * •expansion (1°C) 1.22 X 10-5 0.72 X 10-5 .50 X 10-5

E Modulus or elasticity 8.3 X 106 * 2. 4 X 106 ' * 6.0 X 106(psi)

Poisson's ratio 0.3 0.1 0.3

ST Breaking tensile1960 * 1560 • *stress (psi) 8,500

Sc Breaking compressive * 34000 *stress (psi) 9100 70,000

y Specific 5ur.face 50,000 50,000 2,000energy (ergs/cm2 )

0"\~ * Denotes those propertlesfound by experiment. The source of the

other data was the literature [17], [llJ ror the natural rockand porcelain, respectively.

TABLE 1

3.3 . EXPERIME~TAL EVALUATION OF SOME MATERIAL PROPERTIES

The modulus of elasticity was experimentally evaluated

for granite and marble by placing·electric strain gages

on the top and bottom of beams one inch wide by one inch

deep by twelve inches long .. The beams were loaded at their

quarter points and the strain gage readings recorded. From

this information and simple beam th~ory· for 'the stresses,

the modulus of elasticity was calculated. To evaluate this

property at much higher compressive stresses, the beams

were loaded in axial compression and the strain was

recorded. The average values of the two tests was used

in the analyses.

The tensile strength was ~xperimentally evaluated by

center· point loading of beams one inch wide by one inch

deep by five inches long~ The value of tensile strength

given by such bending tests is usually about twice that

for a uniaxial test [ 3] and this must be taken into

consideration in using the data.

The compressive strength was experimentally evaluated

by axially loading to failure four inch long prism~ with

one inch by one inch square cross-sections.

The coefficient of thermal expansion was found by

measuring the change in length of a 12 inch beam of the

material when it was heated from the freezing point to the

62

boiling point of water. The coefficient calculated from

this range of temperatures was used for a much broader

range of temperatures in analyzing the experiments done

with the laser heat source. This extrapolation is

reasonable since the materi~l does not undergo a phase

change in the· higher temperature of the laser experiments.

The tensile strength of the po~celain balls was

found by applying a compressive load to the ball through".

steel platens.* This loading condition was then analyzed

for the maximum tensile stress in the ball at the time

of failure using the finite element stress ~nalysis program.

The maximum tensile stress is in the interior or the body

as it is for the thermal spalling tests. The size and

shape of the tensile zone is also much the same as for the

thermal spalling condition. For these reasons, the ten-

sile strength found by this method seems particularly

valid to use in the prediction of the spalling response

of the porcelain.

3.4 SPECIFIC EXAMPLES WITH THIN DISKS

3.~.1 General. The analysis technique and the

assumptions or estimates about the. material properties

* This is similar to the Brazilian test for splitting·cylinders.

63

Each of the properties were taken from published

data on ~ater1al which was similar to that used in these

experiments'. The boundary conditions were approximations

based on the known total output of the laser and judgment

of the amount of heat lost from the rock. The amount of

heat lost was considered to be negligible since the tests

are of short durat:t.nn and the rock remains relatively

cool over most of its surface. The cool surface would·'.

not lose much heat by either convection to the air or

by radiation. The fraction of ra~iation energy (of 10.6

micron wavelength) from the carbon dioxide laser, which

the surface would receive, was assumed to be unity. In

all cases, the surface of the rock where the laser was

focused was covered with temperature sensitive paint.'

The analysis of the thermal strains required the use

of the coefficient of the~mal expansi.on and Poisson' 5

ratio in addition to those required for the, temperature

analysis. The coeffic1e~t of thermal expansion was exper-

imentally evaluated over the zero to. one hundred degrees

I "The paint used was "Detectotemp" brand made 'by HardmanInc., Belleville, New Jersey. The painted area at whichthe laser radiation struck the surface would almostimmediately turn black. (This was the color correspondingto the highest temperature indicating capability of thepaint.)

65

centigrade temperature range for samples of rock from

the same .sourc~ as the rock used in the other exper­

iments. The value obtained in this manner was used for

the laser heating experiments in which the temperatures

are, in general, in excess of one hundred degrees cen­

tigrade. Poisson's ratio was experimentally found for

the rock used in the experiments. The range of stresses

used in the.evaluation experiments was about the same

as that encountered in the laser heating experiments.

The application or the analyc~s technique in the

prediction of failure modes involves finding the stresses.

To do this only the modulus of elasticity 1s needed in

addition to those factors which are included in the

analysis or the strains. Since the modulus of elasticity

was found experimentally for the rocks over the same range

of stresses as exist in the thermal spalling experiments,

the accuracy of the analytically predicted stresses can be

expected to be as good as that of the strains.

The modulus of elasticity of th~ porcelain was taken

from published data and thus may not have the accuracy

that is expected in the case of the natural rocks. However,

the porcelain is a man-made material ~n which the quality

control may render the estimate based on published data

quite good.

66

The temperature variation with time on the surface of

thin disks were predicted theoretically by using the

finite element heat 'flow analysis; in c,onjunctlon with

the material properties shown in Table 1. In addition,

the following assumptions were made: 1) The entire

energy output of the laser was absorbed by the rock.

2) There were adiabatic boundary conditions elsewhere

on the. surface. This record was then compared with the

temperatures observed using a radiometric microscope.

In calibrating this instrument, it was found that the

emissivity was about 9~ per cent (a black body is 100

per cent) in an experiment where the surface temperature

was a few hundred degrees centigrade. Details of the

experimental equipment and more data are available [17].

A typical result for these experiments is shown in

Figure 11. The d1fference'between the analytical and

experimental results is usually about ten to twenty per

cent with no pattern of consistency of one being lower

than the other.

The strains caused by heating the center of marble

disks with a laser were compared with strains calculated

analytically. The strains near the edge df the disk were

measured using electric resistaric~ 1train gages and

recorded with an pscilloscope. Multiple gages were used

67

0'\Q::)

300

,.....(.)o.........

~ 200::::> .t:f0::lLIa.~

~--OBSERVED- .-- CALCULATED

TEMPERATURE

2 3TIME (SECONDS)

4 5

FIGURE II: OBSERVED AND CALCULATED TEMPERATURES AT BACK- ..

SIDE CENTERSPOT OF DISC. THIS CONSISTED OF LASING A 0-0511 THICK

MARBLE DISC AT A POWER LEVEL a= 10 WATlS [17]

so that the effects of temperature changes of the gage

would be compensated for. Figure 12 shows both the

analytical and experimental strain as a function of

heating time. The error 1s about ten per cent with the

analytical value lower then the experimental.

3.5 SPECIFIC EXAMPLES WITH CYLINDERS

3.5.1 General. The responses of rock cylinders to

local ~eat1ng was studied both experimentally and analyt­

ically in order to understand the nature of the thermal

spalllng mechanisms. The aprroach taken was to conduct

experiments in which the time required to spall at a

given power level was measured and then .using the analysis

technique to find the temperatures, the stresses and the

releasable strain energy at the time of fracture initiation.

This information allowed a prediction of spalling response

·to be made with the developed theory. This would then be

eompared to the experimental behavior for marble and granite

at different power levels, thus illustrating different

modes of failure.

The specimen shape used to investigate and study the

spalling phenomenon was designed to have ~esponses which

would be similar to that which occurs in the field. The

field cond1tlon is usually a relati,rely large mass of

rock heated over a small portion of its surface. For the

M OF 5L POWER

I

-- EXPERIMENTAL VALUES- - - ANALYTICALLY COMPUTED VALUES

2 3 4 5 6 7' 8 910 \I 12 13

TIME (SECONDS)

LASER SEAWATTS, TOTA

~ ~ ,, ~ ,~~

I MARBLE DISC I

THE DISC IS INSTRUMENTED TO MEASURE RADIAL AND TANGENTIALSTRAINS AT A DISTANCE OF o·6Cl FROM THE ,CENTER, Q\J BOTH SIDES.

FIGURE 12: COMPARISON OF EXPERIMENTAL

AND ANALYTICAL VALUES OF MEMBRANE STRAINS70 [17]

experiments, it was desirable to keep the specimens as

small as possible for ease and economy of preparation

and handling, and it was also necessary that the spec-

imens have an axis of,symmetry to render the two-

dimensional analysis progra~ applicable. The cylindrical

shape satisfied these requirements and was used for the

natural rock (marble and granite) spec1~ens; These

specimens were prepared by drilling cores out of large

blocks which were then cut to the proper length.

Spal11ng experiments were conducted on the granite

and marble cylindrical shaped spe~imens by heating at

the center of one end. A microphone was placed near th~

•heated end to pick up the sound of the spalling which

was a snapping noise audible to the human ear a few feet

away from the specimen. ,The sound of the window or the,

laser being opened was also picked up by the microphone

and when displayed together on an oscilloscope screen

gave a record of the heating time' to cause spall1ng. The

size of the heated spot was controlled by reflecting the

beam from a focusing mirror and adjusting the distance

between the mirror and specimen.

Most specimens were l~ inches long an~ l~ inches

inches in diameter. Initially tests were conducted on

shorter cylinders and it· was concluded that l~ inches of

71

length was sufficient to cause spalling of granite specimens

for the range of heating rates that were used in the exper-

irnents. This length was also easy to handle and position

in the test equipment.

:3.5.2 Variation in Length (Granite). To establish

the minimum length of specimen that could successfully

be used to study spall1ng, experiments were conducted on

granite specimens of different lengths. Specimens 112

1 1inches in diameter with lengths of 2' 1, 12, and 2 inches

long were heated on one end with the laser set at low

power and focused. By using a power in the low end of

the operational range for that laser, the conclusions

reached about the minimum length for spalling also held

true for higher ranges of power rates. This was because

at higher powers the spalling condition was reached in

shorter times, i.e. a smaller region of the material would

be heated thus requiring a smaller amount of cooler

surrounding material to'sufficiently restrain it. The

rocusing apparatus was required in oTder to heat only a

small portion of the surface.1 .

Spalling was not observed in the 2 long specimen; the

1 inch long specimens underwent some spallfng, and the

longer spe~imens exhibited very good spalling. On the

basis of these tests, it was decided that specimens l~

72

inches long would be used for all other tests.

The chips formed by the spalllng were about 3/10

inch in diameter. The front side of the chip, which was

the surface of the specimen, remained the same 1n appear-

ance with no signs of crushing or melting.

For those specimens in which spalling occured, the

temperatures and stresses were calculated with the

analysis technique for the time at which spal11ng was

experimentally observed to start. These temperatures

and stresses are shown in Figure 13and Table 2.

The tensile stresses acting under the heated area

on a plane parallel to the surface were· the largest

tensile stresses in the longer specimen and for the1

specimens shorter than about 2 inch, the largest tensile

stress 1s· the tangential one around the heated spot. Thus,

the first fracture to be initiated in the short specimens

would tend to split the specimen. However) the comb1~ation

of the large fracture area required to completely split the

specimen and the low releasable strain energy associated

with this mode would lend to a stable crack as was observed

in the experiments. After the experiment, the outward

appearance of the specimen was that it was~intact and sound

to the tou~h. The extent of the splitting fracture could

only be estimated because the analysis of the stresses

73

SUMMARY OF EXPERIMENTS AND STRESSES FOR VARIATION IN SPECIMEN LENGTHS

Material. GraniteDiameter = 1.5 inchesPower = 50 wattsLaser = 0.3 inchesTime = 2.0 seconds

No.ot: Maximum Maximum Strength** Energy· Observed Correct

Tests Conditions Tension Compression St Sc Ratio Behavior Predictionks1 ksi ksi ks1

3 0.5" long 1.42 14.8 0.78 34 .5 no spall Yes, butsee be 10''1

2 0.0" long 1.15 15.3 to 15 some spall Yes

3 1.5" long 1.15 15.3 1.56 15 good spall Yes

2 2.0" long 1.15 15.3 15 good spall Yes

-J-t=

NOTE:

* ratio of releasable strain energy to energy required to rormrracture surrace for complete spalling. .

For the ; inch long specimens th~ maximum tensile s~ress was notthe stress required to initiate a spall1ng type rracture but asplitting or the specimen. Estimates show that there is in­su~~icient energy for this fracture to propogate through thespecimen.

** for tensile strength a range of ~ to full value of tensilestrength ~rom beam bending.tests is given.

TABLE 2

1-5

LASER POWER::;: 50 WATTS

LASER BEAM DIAMETER • 0-3 INCH

HEATING TIME • 2-0 SECONDS

".......-rn~..........

ffia:::

-..Jtn

V1

-5

\,,~..-------------~

/¢,~---------------

THE MAXIMUM TENSILE STRESS IN THE VERTICAL DIRECTION .UNDER THE SURFACE HOT SPOT (INDUCES A SPALLING FRACTURE)

THE MAXIMUM TENSILE STRESS IN THE TANGENTIAL DIRECTION- - - IN ANULAR ZONE ABOUT THE SURFACE HOT SPOT (INDUCES

A SPLITTING FRACTURE)

2-0'.1-0 1·5

SPECIMEN LENGTH (INCHES)

0-5o· . . , . ,o

FIGURE 13: STRESSES IN GRANITE CYLINDERS AS

.A FUNCTION OF SPECIMEN LENGTH

after· spI1ttlng was not a two-dimensional symmetric

problem •.

By comparing the maximum calcul~t~d tensile stresses

at the time spall1ng started for the longer specimens, it

is seen that these stresses are about equal to the tensile

strength of the granite.

Investigation of the calculated stress distribution

for the cylinders that underwent spalling at the time of

failure initiation shows that the max~mum compressive

stress at the surface (and parallel to it), where the

heat was applied, was less than the compressive strength

of the granite. This corresponds to the experimental

observations that the outer surface of the chip showed

no signs of crushing or melting.

The amount of energy released by tIle spalling greatly

exceeded that required to .form the new fracture area. The

area of the fracture used for the energy calculation was

based on the experimentally observed chip size.

3.5.3 Variation in Power (Granite). Experiments were

conducted on granite cylinders 11 inches by 11 inches at2' 2

three different power levels using the same diameter laser

beam. The heating time to cause spalling was recorded for

each experiment usi~g a microphone and oscilloscope. The

dat~ for the t~sts are shown in Table 3 and in Figures 14

through 18.

Figure 14 shows the distribution of the temperatures

and the stresses for a given heating time. The general

shape of these functions remains the same throughout the

typically short heating time required to cause the spalling

and for the various power levels. With longer heating

times the temperatures increase in magnitude, penetrate

deepei into the specimen and spread out further over the

surface. The tensile stress which initiates the spall

fracture 1s the Z stress shown along the centerline.

Figure 15 shows the increase in calculated maximum

temperature (on the specimen's centerline at the surface)

with respect to heating time fer the three power levels.

These temperatures are calc~lated on the basis of low

temperature material properties and may not be accurate

for the higher ranges of temperature. Therefore, the'

very high temperatures should be considered only for

illustrative purposes.

Figure 16 shows the variation of the maximum com­

pressive stress and the maximum tensile stress which

causes spalling in the specimen with respec~ to the

heating time. It can be seen that for each power level

the maximum tensile stress exceeds the modulus of rupture

before the maximum compressive stress exceeds the

77

compressive strength.

It 1s to be expected that the material will fail

in tension at a level lower than the modulus of rupture

(which 1s typically from 1.5 to 2.0 times the uniaxial

strength of brittle materials). This is confirmed by

the experimentally observed mean heating time (shown

on the figure) which indicate that the calculated stress

was about ~ of the modulus of rupture.

Thp ratio of the compressive strength to the

modulus of rupture and the ratio of the maximum com­

pressive stress to the maximum tensile stress which

causes spal1ing, are shown in Figure 17 .as a function

of heating time for three different power levels. If

the ratio of the stress~s i~ less than the ratio of the

strengths the tensile strength will be exceeded before .

the compressive strength is. If this is the case, then

a fracture will be initiated which will cause spalling

if there is'sufficient energy available for complete

propagation. If not, the surface will crush, and since

this will subsequently reduce the tensile stresses,· a

spall producing fracture will probably not occur. It is

seen from Figure ·11 that the ratio of stres5es is less

than the rLtio of the strengths, and therefore spalling

can be expected. This prediction was confirmed by

18

experimental observation.

Figure 18 shows the required thermal energy to cause

spall1ng as a function of power. For the granite cylinders

there was a slight decrease observed when increasing the

power from 50 to 200 watts.

In Table 3 the term "Energy Ratio" 1s the ratio of

the strain energy that would be released by complete

propagation to the energy that would be consumed by the

fracture process. Since this term 1s greater than unity

for all three power levels, it was expected that complete

propagation would occur. Experimental observations

corroborated this hypothesis.

3.5.4 Variation in Power (Marble). Experiments were

conducted on marble cylinders l~ inches by l~ inches at

power levels of 50, 100 and 200 watts. Table ~ gives the

details of the .experiments. and parameters. For each· of

these cases (and for a few trial ones at even higher and

lower power 1.evels) there was no sound of spalling recorded

with the microphone and oscilloscope· nor was there any

other evidence that spalling had occured. If the heating

was stopped before melting of the surface began, the heated

material showed signs of crushing. It could be easily

removed wi~h a knife blade.

Figure 19 shows the distribution of temperatures and

79

Granite Cylinders (l~" by l~") Heated with a Laser Beam 0.2 inches in Diameter

Mean(l)Maximum(2) Strength(3No. Heating. Maximum (4~ Agree'"

of Power Time Tension Compression St Sc Energy ~ Pred- . withTests Watts Sec. ks1 ksi ksi ks1 Ratio lction Exp.

4 50 0.83 1.25 13.3 2.9 Tensile Yes.fracture

andspall

.785 100 0.38 1.34 14.0 34. 2.1 Tensile Yes

to fractureand

1.56 spall

5 200 0.18 1.35 15.0 1.5 Tensile Yesfracture

and. spall

ex>o

(1)(2)(3)

(4)

Mean of experimentally observed heating time required to cause first spall.The calculated stresses ~or the mean heating time.The r~nge for the tensile strength is 1/2"to the full value or the modulusor rupture.Ratio or the calculated releasable strain energy to the energy requiredto form the fracture area for complete ral1ure.

TABLE 3

Jes

0::>~

-7-6 ksi -.,-------

695°C

I RADIANT HEAT

V

'!-...JIi. 0.1-....,

1-5"~

¢.. "

STRESS

- - TEMPERATURE

FIGURE 14: TEMPERATURE AND STRESS DISTRIBUTION FOR A

GRANITE CYLINDER (50 WATTS, 0·4 SECONDS HEATING)

5000

.TIME (SECONDS)

5

I

21 •

ANALYTICAL PREDICTION BASED ONLOW TEMPERATURE PROPERTIES

-2

82

IOO-------------_~ _ __L.__ __.a.___._..._ _._.

-I

10,000

FIGURE 15: . MAXIMUM SURFACE TEMPERATURE

VS. HEATING TIME FOR GRANITE CYLINDERS

52

I

~~fJO~

MODULUS OF RUPTURE-----

+-- THE 'CALCULATED MAXIMU~

SPALL PRODUCING TENSILE. STRESS IN THE SPECIMEN

(XlAPRESSIVE STRENGTH-----

r:{.~ THE CALCULATED MAXIMUMfJ 4-- COMPRESSIVE STRESS

IN THE SPECIMEN

11

·2

·1MEAN EXPERIMENTAL HEATING TIME TO SPALL

·5"'--"'_.-.........----....-_..---a__..-.I- .l.-__.-..

·1

TIME (SECONDS)

FIGURE 16: STRESS VS. "HEATING TIME

FOR GRANITE CYLINDERS83

2

50

20

5

en~ " ......,(J) 10t3IX:~ .en

. COMPRESSIVE STRENGTH / TENSILE STRENGTH2.()-~------. ~----.-

---- ---==-~-= .........._.---......----10

oij .. 50:

I

2 STRESS RATIO, 50 WATTS

- • - STRESS RATIO, 100 WATTS

- - - STRESS RATIO, 200 WATTS

. ·5'------...a.----...._-.l"---__~_ __...._'____'~·2 ·5 I 2 5

TIME (SECOr~DS)

.FIGURE 17: VARIATION WITH RESPECT TO

HEATING TIME OF THE RATIO' OF. 'r

MAXIMUM COMPRESSIVE STRESS OVER THE

MAXIMUM SPALL PRODUCING TENSILE STRESS

IN GRANITE CYLINDERS84

I

100 150 200 250 300LASER POWER (WATTS)

GRANITE

~CYLINDERS----------=

PORCELAIN----..,

SPHERES

50o...-.---....Io.-_--a.-_--..-_-.-..... .-....-_-..o

'50

200

~~LLJ..J<t 100

II-

,...

~ 150", .,......

FIGURE 18:· THERMAL ENERGY REQUIRED TO

CAUSE SPALLING VS. POWER LEVEL85

stresses 1n the specimen. This distribution 1s very nearly

like that for the granite cylinders and will c~ange only

in relative size and magnitude with changes 1n heating

time or variation of power levels.

The increase in the calculated maximum temperature

(on the specimen's centerline at the surface) 1s shown

in Figure 20. The analyses were performed based on the

low temperature properties of the" marble and may not be

accurate for the higher temperature ranges with the

accompanying phase changes. Therefore, the highest

temperatures should be considered only for illustrative

purposes.

Figure 21 shows the variation with heating time of

the maximum compressive stress and the maximum tensile .

stress (which would tend'to cause spalling in the

specimen). For each power level the maximum compressive

stress exceeds the compres~ive strength before the

maximum tensile stress exceeds the modulus of rupture.

Thus, it 1s expected that the surface would fail in

compression before a spall" producing fracture could be

initiated. Even if a tensile failure could occur at

one half the modulus of rupture, the compressive failure

at the surface would still be the fIrst mode of failure.

This crushing type of failure was the type that was

86

observed in the experiments.

A crushing of the surface under the laser radiation

has two effects which prevent spalling. First, the

local crushing relieves the spall producing tensile

stresses. Second, the crushing increases the porosity

of the material, which decreases the thermal conductivity.

Thus, a crushed layer will act as an insulator, retaining

the heat at the surface where it would cause melting and

heat 'loss by reradiation. The reduced heat flux into the

intact material underneath would tend to reduce the

thermal gradients and the thermal stresses.

The ratio of the compressive strength to the

modulus of rupture and the ratio of the maximum compressive

stress .to the maximum tensile stress are shown in Figure 22

as a function of heating time for three different power

levels. For all heating times and power levels, the.

ratio of the stresses exceeds the ratio of the strengths

which means that a compressive failure will occur first.

3.5.5 Variation in Power (Porcelain). Porcelain

spheres l~ inches in diameter were heated at three

different power levels and the heating time required to

cause spall1ng was recorded using a microphone and

oscillosco~e. Table 5 gives details of the experiments

and the parameters.

81

1 1Marble Cylinders (1

2" by 1 2 ") Heated With a Laser Beam 0.2 Inches Wide

roco

No. Average (1)strength(2)

Agreeof: Power Stress withTests Watts Ratio Ratio Prediction Experiment

3 50 10.25 11.6 Compressive YesFailure noSpal11ng

4 100 11 LJ.6 Compressive YesFailure noSpal11ng

5 200 9.5 LJ.6 Compressive YesFailure noSpal11ng

(1) Ratio.or maximum compressive stress to the maximum tensile stresswhich tends to induce spalling averaged over the heating time

(2) ~atl0 or the ~ompress1ve strength to the modulus or rupture

TABLE ij

-- STRESS·

- - - TEMPERATURE

"ct.

r1-5"

1

40 ks i..",-- ____

J RADIANT HEAT

1_5"f.4

ex>\D

FIGURE 19: TEMPERATURE AND -STRESS DISTRIBUTION FOR. A

MARBLE CYLINDER. (50 WATTS, 0-6 SECONDS HEAriNG)

5

. I

ANALYTICAL PREDICTION BASED ONLON TEMPERATURE PROPERTIES

-5 I · ~

TIME (SECONDS)

-2 .IOO'-----a..---_....t-_--L-..-_~____.... _____

-I

90

lOP00

FIGURE 20: MAXIMUM SURFACE TEMPERATURE

VS. HEATING TIME FOR MARBLE CYLINDERS

5000

,..".

oe.."2000lIJ. 0:::>

~~ 10001IJI-

5

COMPRESSIVE STRENGTH

THE CALCULATED MAXIMUMCOMPRESSIVE STRESSIN THE SPECIMEN

THE CALCULATED MAXIMUM...-- SR.\LL PRODUCING TENSILE

STRESS IN THE SPECI~1EN

91

MOOUlUS OF RUPTURE------------

FIGURE 21: . STRESS VS. HEATING TIME

FOR MARBLE CYLINDERS

·5~---------._..__ _..._ _..I_______'

·1 . -2 -5 I· 2

TIME (SECONDS)

50

I

92

5-5 I . 2TIME (SECONDS)

·2·5'"-----....&...---__"'----_-..- -.j

·1

2 STRESS RATIO, 50 WAITS

_.- STRESS RATIO, 100 YJATTS

- _.. STRESS RATIO, 200 \VATTS

50r------r----'--T--..-- -r---__---.__----.......

20

oij 5 COMPRESSIVE STRENGTH / TENSILE STRENGTH~ .. -------------------

.FIGURE 22:. VARIATION WITH RESPECT TO

HEATING TIME OF THE RATIO· OF

. MAXIMUM COMPRESSIVE STRESS OVER THE

MAXIMUM SPALL- PRODUCING TENSILE S-rRESS

IN MARBLE CYLINDERS

The distribution of the temperatures and stresses

for a given heating time and power are shown in Figure 23

These are similar ~n pattern to those for the cylinders

and c~ange in the same manner with respect to changes in

power level or heating time.

The variations in calculated temperatures as a function

of time for different power levels are shown in Figure

The analyses were based o~ the materiars low temperature

properties and may not be accurate for the higher tem-

perature ranges. Therefore, the highest temperatures

should be considered for illustrative purposes .only.

The variation of the maximum compressive stress and,

the maximum tensile stress (that causes spalling) with

respect to heating time are shown in Figure 24 for the

three power levels. Observations of the values shows

that the maximum tensile stresses will exceed the

tensile strength well before the maximum compressive

strength for all power levels. The mean experimentally

observed heating times are also depicted in the figure.

The calculated maximum tensile stresses at the~e times

·corresponds well with the tensile strength as found

experimentally by static loads on the spheres.

The ratio of compressive stren~th to the tensile

strength and the ratio of the maximum compressive stress

93

to maximum tensile stress that causes spalling as a

function of the heating time for the three power levels

are shown in ~lgure 25. The ratio-of the stresses is

larger than the ratio of the strengths for all values

of power level and heating time. Thus, the tensile

strength will be exceedeu before the compressive strength

is. This will initiate a fracture and cause a spalling

type of failure if there 1s enough releasable strain

energy for complete propagation.

. Note that the ratio of the stresses was more than

twice that for granite and marble cylinders. This is

because th~ spherical shape used for the porcelain

specimens insured a sharp curvature 1n the compressive

stresses parallel to the surface which cause the

spalling inducing tensile stresses in the vertical

direction to increase. Thus convex surfaces result in

more favorable stress pat~erns for thermal spalling.

Figure 18 shows the required thermal energy to cause

spalling as a function of power. ~or the porcelain

spheres there was a substantial decrease in the required

thermal energy with an increase in· power from'100 to 300

watts. This was not the case with the granite cylinders.

The reason for the difference was the shape of the surface

at the point of neating. For higher powers and the

9~

shor~er heating times required the large compressive

stresses parallel to the surface were much closer to the

surface and constrained to follow it with its relatively

sharp radius.

In general, the porcela.in· required more thermal

energy to cause spalling because the tensile stress required

for fracture initiation in the porcelain was' much higher

than in the granite.",

In Table 5 the term "Energy Ratio" is the ratio

of the strain energy that would be released by a complete

spall at the time of fracture initiation to the energy

that would be absorbed by the fracture process. This

term was greatly in excess of unity for all power levels,

which resulted in very high chip velocity during spallinE

for all experiments.

95

"

1Porcelain Spheres (12

" diameter) Heated with a Laser Beam 0.2 Inches W1.de

Mean (1)

Strength(3;No. Heating Max1mum(2) MaximumEnergy(4)

Agreeo~ Power Time Tension Compression St Sc Pre- withTests Watts Sec. ks1 ks1 ks1 ks1 Rat 1'0 diction Exp.

5 100 1.90 7.8 43.5 17.6 Tensile Yesfracture

andspall

5 200 0.63 9.0 52.0 8.5 70. 14.3 Tensile Yesrracture

andspall

5 300 0.30 9.3 54.0 12.7 Tensile Yes:fracture. andspall

-----/

\.00'\

(1)(2)(3)

(4)

Mean~'or experimentally observed heating time required to cause r1rst spall.The calculated stresses for the mean heating time.The rang~· for the tensile strength 1s 1/2 to the rull value or the modulusof rupture.Ratio or the calculated releasable strain energy to the energy required toform the fracture area :for complete railure.

TABLE 5

~

--.J

~ .1 RADIANT HEM .

-- STRESS- -- TEMPERATURE

t,

FIGURE 23: TEMPERATURE AND STRESS DISTRIBUTION FOR A

PORCELAIN SPHERE' (100 WtuTS, 1·25 SECONDS HEATING)

I

200

ANALYTK;AL PREDICTION BASED ONLOW TEMPERATURE PROPERTIES

100"-------"-----..&0-_-"-__'"""'-- ......--_---..·1 ·2 ·5 I · 2 5

TIME (SECONDS)

5000

IOPOO

FIGURE 24: MAXIMUM SURFACE TEMPERATURE

VS. HEATING TIME FOR PORCELAIN SPHERES'.... ~.

98

5

TENSILE STREN

I

COMPRESSIVE STRENGTH----------

THE CALCULATED MAXIMUM+--COMP.RESSIVE STRESS IN

.THE SPECIMEN

1

~~. ~ +-THE CALCUL~TED MAXIMUM

SPALL PRODUCING TENSILESTRESS IN TliE SPECIMEN

1MEAN EXPERIr.1ENTAL HEATING TIME TO SPALL

FIGURE 25: STRESS VS. HEATING TIME

FOR PORCELAIN SPHERES99

-7

·5------a.----.....--------_~ _.J

-I -2·5 2

TIME (SECONDS)

50

100

2

,...,. .--en~ 10

~

100

2

5-5 I 2

TIME (SECONDS1

-- STRESS RATIO, 100 WATTS l

- • - STRESS RATIO, 200 WATTS

- - - - STRESS RATIO, 300 WATTS

-2

~ . .. (D1PRESSIVE STREt\IGTH /TENSILE STRENGTH--------------

.FIGURE 26: VARIATION WiTH RESPECT TO

HEATING TIME OF THE RATIO OF. ~..,

MAXIMUM COMPRESSIVE STRESS OVER THE

MAXIMUM SPALL PRODUCING TENSILE STRESS

IN PORCELAIN SPHERES

20r-----~---r----,.----~----..-----

o

~

CHAPTER IV

CONCLUSIONS AND FUTURE WORK

ij.l CONCLUSIONS

There are several conclusions that can be made based

upon the results of the analytical and experimental work.

They fall into two main categories: first, the con­

firmation of the analysis technique and second, the under­

standing of the nature of spalllng in brittle materials.

The conclusions,are:

1) It 1s possible to analyze the temperatures and

the resultant thermal stresses with sufficient detail

and accuracy to predict and understand the thermal spalling

response of some brittle materials when locally heated.

2) The analysis technique does not require excessive

computation or precise evaluation of the material prop­

erties and boundary conditions. This is because the

results of the analysis may be checked for accuracy by

comparison with results from simple ~xperiments which

validate the assumptions made about the material properties

and boundary conditions as a group.

3) Thermal spalling by local heating~on the surface

1s due to complete propagation of a fracture initiated by

101

tensile stresses acting below and normal to the heated

surface."

4) The compressive strength of the material must be

sufficiently large c~mpa~ed to the tensile strength to

prevent a compres~ive failure at the surface before the

spall producing fracture can be initiated by a tensile

failure below the surface.

~5) There must be sufficient energy available to form

the new surface area or there will be incomplete fracture

and no spalling. "This was not a problem in the limited

range in which experiments were conducted.

In regard to the above conoluslons, the following

points may be noted with respect to experiments involving

granite, marble and porcelain: The granite and porcelain

specimens spalled at the time the maximum tensile stress

was about equal to the tensile strength. The marble

specimens failed to spall because the surface material

failed in compression before a tensile failure could

initiate a spalling fracture.

4.2 FUTURE ~ORK·

The research conducted for this thes1~ can be

considered as part of a larger research program at the

Massachusetts Institute "of Technology to provide new

102

knowledge which will lead to improved rock excavation

methods. In this thesis, a rational analysis technique

has been developed to evaluate the thermal stresses

which were hypothesized as the basic mechanisms of

thermal spalling. Comparisons have been made, with good

correlation, to data from experiments which covered a

limited range of boundary conditions for a few different

materials.

ror this work to be of greatest possible use in the

development of significant improvements in rock ex­

cavation methods, the analysis techniques and theory

should be applied to experimental results from tests

which have boundary conditions and heat sources similar

to those encountered in the field. Extensions and improve­

ments could then be made) if required, to the technique

and hypotheses for this larger class of problems in the

following areas~

1) Automation of the analysis programs to handle the

moving ·boundary problem. '

2) Analysis of the case of surface crushing in more

detail.

3) The slow growth case where the prqpagation is

dependent on additional heating due to the initial lack

of energy.

103

Based up~n the experience gained during this thesis,

it 1s anticipated that extensions' into the following

related areas will lead to most valuable results in the

understanding and improvement of excavation techniques.

1) The application of. the arlalysis technique to

field type conditions with prototype heat sources, and

the comparison of these results to experimentally recorded

strains: The results of this would indicate the improve­

ments'" required, if any J in the assumptions made about

the boundary conditions and material properties. Then

the analysis technique can be used to study the spalling

in simulated field conditions and improvements in

equipmel1t and/or operating procedure could be done in a

rational manner based on the concept of furnishing the

maximum tensile stress and minimum compressive stress in

the most efficient manner.

2) It is possible to apply the developed analysis

technique to the problem or "how mechanical bits cause

fracture in brittle materials. The ~ost difficult aspect

of this problem is to establish the proper shape and

conditions at the boundary between.the bit and the rock.

At this interface, there 1s crushed materi~l which is

controlling both the magnitude and direction of the

stresses. An experimental study of the properties and

104

flow. of this crushed material would be used to estimate

the proper boundary shape and conditions. Based on these

estimated boundary conditions, analyses could be made of

the stresses and strains. These strains would then be

compared to experimental strains obtained by placing strain

gages on the rock very close to the bit. If necessary, the

estimates of the boundary shape and .conditioris could be

modified in order to achieve the desired close correlation

between the analysis and experiments. When this is sat­

isfactorily done, the analysis technique could be used to

study the initiation and propagation of brittle fracture

caused by mechanical bit action.

3) It has been observed in this work that often

heating alone cannot create a tensile failure in the prQper

location before the material fails in compression on the

surra~e. Also, there are cases in which the available

energy for spalling is sufficient to cause propagation

even though the tensile stresses are insufficient to

initiate the fracture.

For materials 1n which this type of behavior hinders

the spalling rate, a means of initiating the fracture

should be investigated. The most likely means of achieving

this seems to be with mechanical cu~ters. Depending on

the type of Qpera~ion, this could be thought of as a heat

105

assisted mechanical method or a mechanical assisted

thermal spalllng method. The study of this thermal­

mechanical problem accurately and in detail, would

require that the stress problem concerning the inter­

action of the mechanical bit and the rock surface be

done first. Then the method of solution would be to

rationally combine heating and mechanical methods in

the manner which would best initiate the fracture and

propagate it to form chips.

106

REFERENCES

1. Maurer, W., "Novel Drilling Techniques, Pergamon PressLtd., London, 1968.

2. Helseth, H. C. and K01.ller R. H., The Use of Rocket­. Jet Burners, Soc. of Min. Eng. of AIME, Preprint

68-H-325, September 1969.

3. Fairhurst, C., Failure and Breakage of Rock, EightSy~posium on Rock Mechanics Port City Press, Inc.,Baltimore, Md. P. 335, 1967.

4. Evans, I. and Murrell, S. A. F., Wedge Penetration'into Coal, Colliery Engineer, 1962, Vol. 39, No. 455.

5. Paul, B. and Sikarskie, D. L., A Preliminary Theoryon Static Penetration by a Rigid Wedge into aBrittle Material, AIME, Transactions, 1965, Vol. 232.

6. Garner, N. E., The Photoelastlc Determination of theStress Distribution Caused by a Bit Tooth on anIndexed Surface, M. S. Thesis, Univ. of Texas,January 1961.

7. Norton, F. H., A General Theory of SpallingJ. Am. Cerami Soc. V. 8, 1925, PP 29-39.

8. Norton, F. H., The Mechanism of SpallingJ. Am. CeramI Soc., V.9, 1926, PP 446-461.

9. Norton, F. H., Discussion on Theory of SpallingJ. Am. Cerami Soc., V. 16,1933, PP 423-424.

10. Preston, F. W., Theory of Spalling, J. Am. CeramiSoc., V. 16,1933, PP 131-133.'

11. Kingery, W. D., Factors Affect1.ng Thermal StressResistance of Ceramic Materials, J. Am. CeramiSoc., v. 38, No.1, January 1; 1955.

12 .. ~asselman, D. P. H., Elastic Energy at Fractureand Su~face Energy as Design Criteria for ThermalShock, J. Am. CeramI Soc., V. ~6, No.ll, November 1963.

REFERENCES(Continued)

13. Griffith, A. A., The Phenomena of Rupture andFlow in Solids, Phil. Trans. Rog. Soc., 1921,Vol. A 221, PP 163-198.

14. Tetelman, A. S. and McEvily,- A.. J., Fracture ofStructural Materials, John Wiley and Sons Inc.,New York, 1967.

15. Paulding, B. W., Crack Growth During BrittleFracture in Compression, Ph. D. Thesis, June~965, Massachusetts Institute of Technology.

16. Rad, P. F., Personal Communication, Departmentof Civil Engineering, Materials Division,Massachusetts Institute of Technology.

17. Farra, G., Nelson, C. R., Moavenzadeh, F.,Experimental Observations of Rock Failuredue to Laser Radiation, MassachusettsInstitute of Technology Research Report R69-16,December 1968.

18. Z1enkiewicz, O. C., and Cheuney, Y. K., The FiniteElement Method in Structural and Contenum Mechanics,McGraw-Hill Publishing Co., London 1967.

19. Fuj1no, T., Ohsaka, K., The Heat Conduction andThermal Stress Analysis by the Finite Element Method,Technical Headquarter, Mitsubishi Heavy Industries,Ltd.) Tokyo, Japan.

20. Wilson, E.L., A Digital Computer Program for the FiniteElement Analysis of Solids with Nonlinear MaterialProperties, Technical Memorandum No. 23, Aerojet­General Corporation, July 1965.

21. Carslaw, H. S.. , and J. C. Jaeg~r, Conduction of Heatin Solids, Oxford University Press, New York, 2nd

. ed., 1959.

108

BIOGRAPHY OF THE AUTHOR

Name: Charles R. Nelson

Born: September 23, 1940, in Grafton, North Dakota

Education:

Secondary Schools in Grafton, N9rth Dak6ta

Sept. 1958 to June 1962: University of North Dakota

Sept. 1962 to present: Massachusetts Institute ofTechnology

Received the degree of Bachelor of Science from theUniversity of North Dakota in 1962, the degree ofMaster of Science from the Massachusetts Instituteof Technology in 1965 and the degree of'CivilEngineer in 1967.

Professional Experience:

June 1967 - Jan. 1969: Staff Engineer for Simpson,Gumpertz &Heger Inc., Consulting Engineers.Cambridge, Massachusetts.

Sept. 1966 - June 1967: Instructor in the CivilEngineering Department at the MassachusettsInstitute of Technology. Cambridge, Mass.

June 1964 - Sept. 1964: Engineer for LincolnLaboratory, Lexington, Massachusetts.

Sept. 1962 - June 196~: Full-time Research Assistantin the CiVil Engineering Department at theMassachusetts Institute of Tech~ology. Cambridge,Massac.husetts.

Honors:

The author graduated Cum Laude from the University ofNorth Dakota and received the Outstanding CivilEngineer A'l1ard of 1962", He was a National Science~Fellow from 1964 to 1966,

'lOg

BIOGRAPHY OF THE AUTHOR(Continued)

Publications:

Massachusetts Institute of Technology Report,;R64-37 "The Dynamic Response of Reinfurced:Concrete Circular Arches".

Massachusetts Institute of Technology ReportR69-16 "Experimental Observations of RockFailure due to Laser Radiation~'.

The author was married to Marlys Brown in 1959and 1s the father of two children, Klmberleeand Brent.

. .The author 1s joining the School of Minerals andMetallurgical Engineering at the University ofMinnesota as an Assistant Professor of Geo­Engineering.

110

APPENDIX A

HEAT TRANSFER BY THE FINITE ELEMENT METHOD

,In th~s method the body 1s considered divided into .

small finite sized elements which are kept compatible

with each other and also satisfy overall thermal equl1ib-

rium. The problem 1s thus transformed from solving

a differential equation with boundary conditions to one

of solving ordinary simultaneous equations.~

The finite element method of heat analysis developed

here will handle most types and shapes' of linear bound-

ary conditions. Nonhomogeneous and nonlsotropic rocks

can also be handled, but the inhomogeneity must vary

slowly when compared to the finite element size. The

nonlinear aspects of the problem may be handled by u~ing

the incremental loading approach. Using this approach

the load is applied in small increments such that linear

boundary conditions and material properties may be used.

during that increment of load. At the end of each load

increment, new boundary conditions and material properties

are calculated for the next increment of load.

For the finite element method the size and the complex-

ity of the problem that can be handled. is limited by

III

the amount of computation time available. There are two

classes of pro~lems which require a much reduced number

of calculations and still provide solutions useful in

many types of real problems. The first class 1s that of

plane strain. This condition is approximated in thin

slabs. The other class 1s the ax1-symmetrlc one in which

all boundary conditions, loadings, and material properties

.3.re symmetric to a single axis. It 1s the type of prob lem

'chat"exists in discs and cylinders. By virtue of the

symmetry in such a case, only one pie-shaped wedge need be

considered. This effectively reduces the three-dimensional

problem to a two-dimensional one.

A method to evaluate heat transfer by the finite

element technique for rock specimens subjected to laser

radiation has been developed in the following manner:

In general the element shape in this work 1s tri­

angular and the temperature within the eJement is assumed

to be a linear function of the x and y coordinates:

T = A + Bx + Cy

The values of A, B, and C must be ex~ressed in terms

of the temperatures· at" the three corners of the element

called Ti' T2 , and T 3 •·

112

adX

(TK )y

a-3Y

113

(TK )t dx +y

aelY

and j 1s:

The heat flow of the edge of an ele~ent between points i

where Kx and Ky a?e the conductivities of the material

in the x and y dlrectio~ respectively.

and in the y direction:

1 Xl YI A

= 1 X2 Y2 B

1 x 3 Y3 C

A 1 Xl Yl-1

T 1

B = 1 x2 Y2

C 1 X3 Y3

For a given element, the heat flow intensity in the x

direction is:

qx - - a (TK )n- x

element, the intensities become:

Y, one finds:

K aTYdY

K aTxdX

ll~

qy - - Ky C

q - - K Bx x

qx

k

....-.------ x

y

The values of B .and C in terms of T1 " T2

, and T3

are:

on substituting for T in terms of A, B, C, X and

iIf the value of conductivity does not vary within an

That 1s

1s the area of the element

are the coordinates of the

1 xi YiA = 1 1 xj Yj2

1 xk Yk

rj q dy + rk q dy + JY i q dy = 0Yk x Y

jx Y

kx

115

an element of either q or q would be equal to zero.x y

It should be noted that, if the element were of con­

stant thickness, an integration around all three sides of

which 1s

where xi' Yi ' xj ' xk ' Yknodes of the element and A

Therefore the element, by itself, 1s in thermal equilibrium.

However, along a boundary between two elements there will

be, in general, a discontinuity in the heat flow intensity.

This discontinuity will be assumed to cause the temperatllre

along this boundary to change. This is expressed analyt­

ically in the following way. The heat flow out of an

element 1s assumed concentrated at the ~earest node. Thus

the heat into node i 1s:

IXx j IYtq dx + 1 j tq dy

Y 2 Y x1 1

•The total heat flow into node 1 would be found by summing

over all the elements containing node i :

Q = I: Qitotal

If node i changes in temperature, all elements containing

node 1 must change their internal temperatur~ distrib-

utions in a linear fashion. This follows from the initial

"assumption that the temperature is.at all times a linear

function of the space coordinates.. .

It is now possible to find·the heat required to change

the'temperature of only.one node of an element. This term

117

change 1s as shown:

T[(x Y -x y ) + (y -y )x +k j j k k j

IjCP = tp(SpH)

(X -x )y]dxdy = 1 tp(SpH)TAj k 3

element---

------ temperaturefunction

Thus the heat required to raise only one node to a given

temperature, with. a linear distribution within the ele-'f

ment, is 1 that required to raise the entire element to3

that temperature. Since· there 1s, in general) more than

by T 1s:

the heat required to change the temperature of node i

Assuming that the thickness (t), ~enslty (p), and

specific heat (SpH) are constants within the element,

1s analogous to a structural stiffness. The temperature

118

at a riode of:

Summing over the several elements at a node:

= 1 tA (HG)3

HG = Heat Generated per Unit Volume

HGN

·HGN = EHGNtotal

CPT = E CP

Boundary conditions at any point may pe specified..

in one or more of the following ways: given temperature

(e.g., water bath), given heat flux (e.g., laser), ra-

the nodes.

heat capacity of each element was divided equally among

equally to each node of the element in the same way as the

it would uniformly raise the temperature of that element.

Th1s~·wil1 be accounted for by having this heat divided

If there is any heat being generated within the element,

all adjacent elements giving a total heat capacity (CPT). .

one element at a node, the effect must be summed over

given temperatures.

at the nearest boundary nodes.

of elements to account for the conductivity of the

aT CPT = aat

lig

means differentiation with respect to time.a

at

Qtotal + Qboundary + HGNtotal -

written as :

Tboundary = Tgiven

Qboundary = Jf qboundary,

boun,dary condi tlon.

The heat equilibrium equation at each node can now be

The conductive zone is handled by adding an extra layer

The ~~diation and flux conditions are assumed concentrated

dlat.ioI1, or a ,conductive zone to some given temperature.

the boundary nodal temperatures are held fixed at those

as a set of given temperatures is "easy to analyze since

The case in which the boundary conditions are specified

'boundary layer and thus converting it to a fixed temperature

the system would be equal to the total number of nodes

120

Qtotal + Qboundary + HGNtotalaT CPT =at

If we let:

~hus, at this point, there are N equations, NaTunknown temperatures, and N unknown --, that 1s, aat

total of 2N unknowns. If the steady state solution 1saT .

wanted, the N unknown at' s are zero. .The system now

has N unknowns and N equations which can be solved.

temperature was given would be nUll.

would be the same as the number of unknown temperatures

since the equations for the boundary nodes where the

perature was given. The number of independent equations

minus the number of boundary nodes at which the tem-

An equation of this nature can be written for each node

in the system. The number of unknown temperatures in

. We get:

The temperature in the future, Tf , is predicted in terms

of the present temperature, Tp ' and the time increment,

6t, into the future. The Qtotal' Qboundary' and

HGNtotal are functions of the present temperatures and

known boundary conditions. The temperature for all

future time can now be found by marching the solution

out from some known initial boundary conditions.

The maximum rate at which the solution can be

marched forward in time 1s controlled by the stability

of the sOlution. To find the maximum rate, the following

logic 1s used:

Assume there is an error introduced into the solution

at some point due to round off or other causes. If this

does not diminish with time, the solution will become unsta-

ble. If the system had zero values throughout except for

an error (E) at one node, then this error should diminish

with time. Initially, Ti

= E, all other T = 0 and the

heat flow into node i is Qtotal + Qboundary' After one

time increment At the temperature at ·node 1 1s:

Q + QT (At) = (total boundarY)At + E

1 CPT

121

Now if:'

IT (~t)1 > IT (0·)1i 1

there 1s instability since the error may grow with time.

For the limiting case:

IT (l1t) I = IT (0) I1 1

which gives:

Qtotal + Qboundar~CPT ~t = 0, -2E

For the first case a lower limit on ~t 1s zero, that

is, in a heat transfer problem one can not predict back

in time. The second case will give the maximum rate

at which the solution can be marched forward. Neglecting

the Q (boundary) term:

Qtotal--- ~t = -2E

CPT

122

one dimension the condition 1s:

have zero temperature and the problem 1s linear, E can

x1+2

2CPT

Qtotal(l)

~t = -2

123

-------,-E

o < ~t <

Qtotal(l)CPT

i-2

T

where Qtotal(l) 1s the Q of that node which has a unit

temperature and all other nodes have zero temperature. In

be factored out of Q After cancellation of Etotal-

there 1s for the limiting case:

ones adjacent to it. Since, initially, the adjacent nodes

node 'to another.' The Q 1s a function of these astotal

well as the temperature of the node in question and the

arrangement and material properties, and varies from one

CPT is a function of the element size, shape, thickness,

If the initial conditions are as shown, the time in­

crement (~t) must be limited such that after (~t),

T. is less than E but greater than -E.1

Note that the minimum v~lue of ~t must be used

after checking all nodes and also that CPT grows with

element size, and Qtotal (1) diminishes with element

size. Thus smaller elements require smaller time

increments."

The computer program to do the heat transfer was

checked by considering a problem which was solved in

closed form [21J. The problem consisted of a wall at

constant temperature which, at time zero, had its out­

side surface temperature changed to zero. For the

computer solution the wall was'split (about its axis of

symmetry) and one half of it was represented by twenty

elements such as:

= 0

o

•

12~

\

The computer solution was within 0.1 per cent of the plots

for the closed form s~lution given by Carslaw and Jaeger.[21].

In varying the time increment in this p~oblem, the derived

limit for stability was established.

125

APPENDIX B

(Approximate)

~ (0 a +0 a +0 a )E x Y y z z x

126.

PE =

=

+

strain energy per unit volume is then:

Where Vi 1s the volume of the element. The total

An approximation of the strain energy in the body can

be obtained by assuming that for the case of suddent

It was necessary to calculate the strain energy in

STRAIN ENERGY CALCULATION

N

PE = ~ PEi1=1

where N is the total number of elements.

energy is then

in element 1 is [ ]:

the specimens before and ar~er the formation of a spall.

finite element over all of the elements. The strain energy

This was done by summing the strain energy in each

._ heating at .the surface the strain is limi ted to one

dimension (in the direction normal to the surface). The

where AT is the change in temperature applied to a single

element 1s:

where ~Ti 1s the change in temperature of the element.

The total energy is:

N

PE =~ PE11=1

The comparison of this approximation with the exact

value shows that for the cases investigated in this work

the approximation is about four times the exact value.

This means that displacement parallel to the surface

reduces the stresses by about a factor of two from the

stresses that would exist if the problem were truly one­

dimensional. The more precise method requires that the

stresses be evaluated while the second method only

required knowledge of the temperature~.

127

APPENDIX C .

DETAILED INSTRUCTION FOR "J.HE ANALYSIS TECHNIQUE

The required modeling and data preparation for the

analysis technique will be shown for a thermal stress

problem.

First, to evaluate the temperature distribution,

the appropriate boundary conditions and material properties

must ··be known or estimated. For granite the following

estimates of the properties are based on published data [ ]

Conductivity = 1./(31500 + 21.6T)BTU in sec of

Specific Heat = .21 BTU/lb of

Density = .094 Ibs/1n3

The boundary conditions are the estimated heat flow

into and out of the material. For this example the heat

flow into or out of the boundary of the specimen at all

places other than that area heated by the laser are

assumed to be zero. This assumption has been proven

valid for short heating times with the laser in which

the surface does not become incandescent.

128

It is estimated that the total energy of the laser

(as recorded by· absorption in the power measuring

calorimeter)' is .absorbed by the rock. This estimate

appears to be valid because the surface has several

impurities, such as water and dus~, and has irregularities,

such as micro-cracks and adhesions all of which tend to

increase the surface absorptivity. The estimated shape

of the beam 1s that of an axi-symmetric gaussian curve.

Thes~' estimates are accurate if the laser 1s operating in

a single mode, which 1s normally sought in operation for

accurate targeting of the beam.

The specimen to be analyzed must be modeled by a

finite number of triangular elements. Since the

temperature can only vary in a linear manner in any element,

each element must be sufficiently small such that the true

shape of the temperature function can be well estimated

by the small straight segments.

For the example under consideration the shape is a

cylinder) 112 inches long and 11 inches in diameter. Of2 ·

this, only the top center zone which ~ill experience an

increase in temperature need be modeled by the· finite

elements. That part of the body which re~ains at a

constant telnperature does not affect the solution at its·

accuracy.

129

The mode~ used for this example 1s shown in Figure 27.

Note that due consideration has been given to the axi­

symmetry of the problem.

The input for the computer program is shown in the

listing. The first line,of data input has the

following meanings.

TERM

KTMAX

KTPRT

NODES

NOEL

VALUE

200

10

53

79

130

. PHYSICAL MEANING

The total number of time

steps the program will

take.

The number .of time steps

the program takes between

printing of the tem­

peratures.

The exact number of nodes

(joints) there are in the

model.

The exact number of el­

ements there are in the

model.

TERM

DENS

SPHET

DTIME

COND

BTEMP

VALUE

.094

.21

.02

1./(31500+2l.6T)

o.

PHYSICAL MEANING

The density of the material

(lbs./in3) •

The specific heat of the

materl~l (B.T.D./lb.)

The step size in time the

program takes in finding

new temperatures.

The thermal conductivity

of the material (B.T.D./

in. ~F sec.) (this input

1s overridden by the non-

linear function for the

conductivity in the

program. )

The temperature any bound-,

ary may be fixed at. (not

needed for the- example) (OR)

tory

~

1.

...,

47, x~ i 9_. . . =" ~ " -,

~

Wf\.)

21 UNDERLINED NUMBERS- ARE ELEMENT NUMBERS

FIGURE 27: GRID FOR HEAT TRANSFER (SEE FIGURE 29 .FOR FINE GRID)

The next group of input which 1s labeled "nodal

data" is established ill the following manner.

The first term is the radial coordinate in inches

and the second 1s the coordinate in the vertical direction

in inches. The third term 1s the initial temperature

of the body at that node. For this example the bodyowas estimated to be at 70 F initially. The fourth term

1s the net heat flow into the node 1n B.T.U. per second

due to outside sources such as the laser. It 1s zero

except for those nodes at the surface heated by the

laser. This heat flow per node is equal to that heat

which 1s incident on the surface which" ~s closest to

the node in question. The fifth term denotes if that

node is a boundary node which must be held fixed at the

specified boundary temperature. If th"1s term 1s negative,

the node 1s not held to the boundary temperature. It

should be made positive if it is desired to fix the node

at the given boundary temperature such as would be the

approximate case for a water bath for example. For the

laser heating example, this term should be negative for

all the nodes.

The next group of input data is the information which:.

tells the computer how to connect the nodes with elements.

133

Each element should have the three nodes associated with

it listed in the same order as the coordinate system. That

is, number the nodes counter-~lockwise in a right-handed

coordinate system and clockwise in a left-handed coordinate

system.

This ends the required input. The rest of the output

1s data that the program generates and 1s printed to help

in checklng the input for error. The next group of print-

out.are the number of the elements which touch each node.

A rapid comparison~ of this with the figure will indicate

if an error has been made in the input of the element

nodes.

The next·group is the area of each element printed

in numerical order. Again, a comparison of this data

with the figure looking for elements with common areas

is a quick way to check the accuracy of the input.

Following this is data on the total specific heat

of each node. It has little value as a check of input

data, but is included as a possible aid in debugging the

program if changes to it are ever made.

All of the data printed out from this point is the

temperatures of t~e nodes after a fixed integer number of~

time steps forward in time. The term KSTOP is the number

of time increments whic~ have posted before the temperatures

are reached. This can be converted to the amount of heating,

134

time by multiplying the KSTOP by the DTIME. Th~ DTIME

1s the size of" the increment in seconds.

Following each print of the KSTOP is the temperature

of each node listed in numerical order.

Once the temperatures have been. calculated over the

range of times of interest, it 1s necessary to calculate

the thermal stresses for specific times. For the thermal

stress analysis, the body 1s modeled by many finite

elements as it was for the heat transfer analysis. However,

there are two modifications required for the thermal stress

grid. One is that the size of the elements may have to be

of a different size for sufficient accuracy. Second, the

whole body must be modeled because the stresses in the

heated zone are influenced by the restraining action of the

material which remains at room temperature. With this in

mind the body is modeled by the grid shown in Figure 29.

The input for this problem 1s shown in listing

Following this is the information on the stress. analysis

program called "Feast In distribute~ by Professor John

Christian of the Soils Division of the Civil Engineering

Department at the Massachusetts Institute of Technology.

Professor Christian is in charge of the implementation

135

of this computer program at the r~assachusetts Institute of

Technology's Computation Center. This information explains

1n detail the input and output of the program for thermal

stresses.

;For the example under consideration, the use of the

temperatures from the developed heat transfer computer

program in the stress analysis computer program requires

interpolation of the data because the grid size 1s smaller

in the stress analysis program. This interpolation can

be done 1n a linear manner because the heat transfer

a~alysis program is based on the assumption that the

temperature is a linear function across each element.

One apparent difference between the two computer

programs is the use of quadrilateral elements along with

triangular ones in the stress analysis. This presents

no problem whatsoever because the only information exchange

between the two programs is the nodal temperatures and no

information which requires an element correspondence. See

Figure 29 for the model used in this example.

136

~. tR

...I , I I I I , 'fL...., JQ" I" '" ">I ." 2. -+

~

23 , 22 I 11 I ~ I !l .~

47 iR Se'l '9 ' 1)1 9 i ~ .• I I i I I i

14 ~

& !Q.

J-'

, I15 • .2- -LA)

--J

14 9- I6

21 UNDERLINED NUMBERSARE ELEMENT NUMBERS

FIGURE 28: GRID FOR STRESS ANALYSIS (SEE FIGURE 29 FOR· FINE GRIP)

. 30 ~

2840

501 6B\ "I 01 \ '191 uVl 'tIl -71

1'.

84 75 69 63 57

..§§. .§Q ~ 4~ 42J 30

51 79 44 70 39 ~A 3 27

1--' I 76 /1 §J! / 160 /°1 :&. / 173 67 !!. M

lA>ex> 8 76 71 64 59 5585 M

:a 68 62 56 50

52 81 45 72 40

75 69 §. 57

86 82 77

76 70 64 58 '52

53V 46V 41V ;z+1! ........, 53 83 46 1.! 41 62

HEAT TRANSFER GRIDSTRESS ANALYSIS GRID

FIGURE 29: FINE GRID FOR FINITE ELEMENT PROGRAMS

PROGRAM "FEAST 1 - 65"

USER'S MANUAL

DATE: MAY 1967

LANGUAGE FORTRAN IV G LEVEL

PROGRAMMER: E. L. WILSON (UNIV. OF CALIF. 1966)

MODIFIED: JOHN T. CHRISTIAN, M.I.T.

DESCRIPTION: The purpose of this computer program 1s to

determine" the deformations and stresses within certain

types of stressed bodies. The program will analyze problem

situations in any of the following cat~gories:

Axial symmetry

Plane stress

Plane strain

Elastic, non-linear material properties are considered by

a successive approximation technique. The effects of dis­

placement or stress boundary condition, concentrated loads,

gravity forces and temperature changes are included.

PROGRAM CAPABILITIES: The following restrictions are

placed on the size of problems which can "be handled by the

program.

139

Item Maximum Number

Nodal Points goo

Ele~ents 800

Materials 12

Boundary Pressure Cards 200

The program incorporates a data-gen~rating facility where­

by only a minimum amount of information neeo. be inputted

to specify the problem topology and geometries. Use of

this capability is described in the next section. The

program permits the use of quadrilateral and triangular

elements, as well as skew boundaries. If specified, a

data deck will be punched to provide the source data for

STRESSPLOT. This plots the vectors of major and minor

principal stress for each element.

Printed output includes:

1. Reprint of Input Data

2. Nodal Point Displacements

3. Stresses at the center of each element.

4. An approximate fundamental frequency. (The

displacements for the· given load condition

are used as an approximate mode shape in the

calculation of a frequency by Rayleigh's pro­

cedure. A considerable amount of engineering

judgement must be used in the interpretation

. of this frequency.)

140

Running time for typical problems 1s about 10 mins.

INPUT DATA FORMAT:

A. IDENTIFICATION CARD -(20A4)

Columns 1 to 80 of this card contain information

to be printed with results.

B. CONTROL CARD - (415, 3F10.2, 315)

,

Columns 1 - 5 Number of nodal points

6 10 Number of elements

11 - 15 Number pf different materials

16 20 Number of boundary pressure

cards

21 - 30 Axial acceleration in the Z-

direction

31 - 40 Angular velocity

41 - 50 Reference temperature (stress

free temperature)

51 - 55 Number of approximations

56 60 = 0 A~lsymmetr1c analysis

= 1 Plane stress analysis

= -1 Plane strain'

65 = 1 A deck of stresses and

di~placements will be

punched.

141

C. MATERIAL PROPERTY INFORMATION

First Card - (215, 2FIO.O)

•Following Cards - (8FIO.O) One card for each temper-

Poisson's ratio - vrz

Modulus.of elasticity - EePoisson's ratio ver and vezCoefficient of thermal ex-

elastic modulus .

for which properties are given =

8 maximum.

pansion as

80 Yield stress 0y

any number 1 to 12.

6 - 10 Number of different temperatures

1 - 5 Materials identification -

ure

1 - 10 Temperature

71

pans ion - a and ar z61 - 70 Coeffic1~nt of thermal ex-

11 - 20 Modulus of elasticity - Erand Ez

11 - 20 Mass density of material

21 - 30 Ratio of plastic modulus to

21 ~ 30

. 31 - 40

41 - 50

51 - 60

Columns

Columns

The following group of cards must be supplied

for each different material:

,

Material properties vs. temperature are input

for each material in tabular form. The properties for

each element in the system are then evaluated by inter­

polation. The mass density of the material is required

only if acceleration loads are specified or if the

approximate frequency 1s desired. Listing of the co­

efficients of thermal expansion are necessary only for

thermal s~ress analysis. The plastic modulils ratio and

the yield stress are specified only if nonlinear materials

are used.

D. NODAL POINT CARDS - (215, 5FIO.D)

One card for each nodal point with the following

information:

Columns 1 - 5 Nodal point number

10 Number which indicates if dis­

placements or forces are to be

specified

11 - 20 R - ordinate

21 - 30 z - ord~nate

31 - ~O XR

41 - 50 XR

· 51 - 60 Temperature

If the number in column 10 is

Condition

0 XR 1s the specified R-load and

XZ 1s the specified Z-load. free

1 XR 1s the specified R-displacement and

XZ ls the specified Z-load.

2 XR 1s the specified R-load and

XZ is the specified Z-displacement.

3 XR is the specified R-displacement and

XZ 1s the specified Z-displacement. fixed

All loads are considered to be total f6rces acting

on a one radian segment (or Ulli t thickness in the case

of plane stress analysis). Nodal point cards must be in

'numerical sequence. If cards are omi tted J the onli tted

nodal points are generated at equal intervals along a

straight line between the defined nodal points. The neces-

sary temperatures are determined by linear interpolation.

The boundary code (column 10), XR a~d XZ are set equal to

zero.

SKEW BOUNDARIES

If the number in columns 6-10 of the nodal point

cards is other than 0, 1, 2 or 3, it is interpreted as

the magnitude of an angle in degrees. This angle is,shown on the following page.

1~4

s

One card for each element ..Element number

145

6 10 Nodal Point I

1 - 5

21 - .25 Nodal Point L

11 - 15 Nodal Point J

16 - 20 Nodal Point K

XR 1s the specified load in the s-dlrect1on

XZ is the specified displacement in the n­

direction

Columns

E. ELEMENT CARDS - (6I5)

1s the same as -179.0 degrees. The displacements of these

nodal po~nts whi.ch are printed by the program are

ur = the displacement in the s-directlon

Uz = the displacement in the n-direction

The angle must always be input as a negative angle and

may range from -.001 to -180 degrees. Hence, +1.0 degree

rThe terms in columns 31 - 50 of the nodal point card

are then interpreted as follows:

26 - 30 Material Identification

1. Order nodal points counter-clockwise around

element.

2. Maximum difference between nodal point I.D.

must be less than 50.

Element cards must be in element number sequence. If

e'lement calads are omitted, the program automatically

generates the omitted information by incrementing by one

the' preceding I, J, K and L. The material identification

code for the generated cards is set equal to the value

given on the last card. The last element card must

always be supplied.

Triangular elements are also permissible; they are

identified by repeating the last nodal point number (i.e.,

I, J, K, L).·

. F. PRESSURE CARDS ~ (215, IFIO.O)

One card for each boundary element which is

subjected to a normal pressure.

Columns 1 - 5 Nodal Point I

6 - 10 Nodal Point J

11 - 20 Normal Pressure

Pressure

As shown previously, the boundary element must be

on the left as one progresses from I. to J. Surface

tensile force is input as a negative pressure.

PROGRAM USE:

a). USE OF THE PLANE STRESS OPTION

A one punch in column 60 of the control card

indicates the body is a plane stress structure of unit

thickness. In the case of plane stress analysis, the

material property cards are interpreted as follows:

Column 11 - 20 Modulus of elasticity Er

21 30 Poisson's ratio•31 - ~O Modulus of elasticity Ez

The corresponding stress-strain relationship

·used in the analysis is

a lJ 'V 0 ~rr

E£

" = rX v 1 0 zz -Z

ll-'J2

0lJ-~

2(lJ+v)

where

ErEz

147

b). USE OF THE PLANE STRAIN OPTION

A punch of -1 in column 60 of the central card

indicates that the body is a plane strain structure. The

material property cards are interpreted as follows:

Column 11 - 20 Modulus of Elasticity Ev21 - 30 Poisson's Ratio v

vH31 - 40 Modulus of Elasticity E

H41 50 Poisson's Ratio 'J

HH

The program must be run on the IBM 360(65 with the as

control cards. Two scratch tapes must be mounted on

logical units 8 and 9. The source program has been

punched on the 026 key punch (BCD).

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KSrnp= 10~1.11tl0 21.1111(' 21.11110· 11.1111021.11110 11.11110 21. 1111 C 21.1111~11.tlllC ?J.• l111t1 ·~l.tl'lr '1.1111021.1111C ll.1lll0 2\.1111" 21.11110'I.tltle ll.11110 21.11110 ?l.lllIC21.11110 21.1111" 21.11110 21.1111021.1111C 21.1l11" 11.11111 21.1111121.11111 21.11110 21.11116 21.11412~1.11S.l4 21.11SS1 71.11110 21.1111021.11113 21. 115'.1 21.21647 21.3l·JQd11.1&;767 21.1124' ·21.21107 21.6Q9?O29.81Q37 30.9')134 ~1.111l~ 21.1111021.t7C;f)~ 2'».'ilPS') 2tlO.2Q?4R 183.60327388.44~7q

KSTnI':: 2021.1111(' 21.1111' 21.11!lC 21.1111011.11110 '1.11111') 21.1111(\ 2 1• 1111021.tlll~ 71.11110 21.11110 21.1111011.1111~ 21.t1!lf) 21.1111" 21.11110~1.1111C 21.1111(" 21.1111C 21.1111021.1111" 2l.lllltJ 21.1111C 21.1111011.1111(\ 21.1ll1l 21.11ZSQ 21. 1123221.11318 ?1.1tll1 11.11444 21.1644921.1~2A2 21.1Q2eJ6 11.11110 21.11110~1.11l17 ?1.184l.17 2~.11-'14 22. 11 qq2l2.QS30' 21.13'66 27.2'3378 4l.99t)b9IiO.OlR77 54.4~17" 21.1111f' 21.1113b21.56112 35.82C9S 5C4.1'32~8 ~8~.31193

6Q4. AS 11 RK STOO= '4t}

21.1111(' 2"I. 11 11 0 21.lll1r 21.11110'21.11110 j)1.1111':' 21.11110 21.1111j21.1111" ?\.\1110 11.1111(\ 21.1111021.1111(\ 21.1111Ct 11.11110 21.1111011.11110 '1.1\\10 21.11110 21.1\11021.1111' 21.11118 21.1111C 21.11110'1.1111(\ 11. t J17'1 21.12'lr 21.12(\4221.tle8' 21.11185 21.1:\423 21.3451821.4?772 '1.413?A 11.1\110 21.1111521 • l' 555 21.4254!l 24.48441 25.6330126.1~h17 71.:?1104 24.110Q3 61.~,qr;815.02322 8 't. 141 86 • ll.II110 21. 1124922.3'400 4A.4liR74 6R3.Q3<J70 q~1.3~200

9b2.8Q01AIt: STt1P= itO

21.1111C 21.11110 21.1111C 21.1111021.1111(' ~1.1111t) 21.11110 21.1111021.1111C 21.11110 11.11110 21.1111021.11110' 21.lltl0 21.11110 21.1111021.11110 21.11110 'l.lllle, 21.11! 1021.1115C 21.11111• ll.1111C 21.1111021.t1112 21.11 1t65 21.1511'i 21. 1463621.18'4~ ?1.11494 21. tqlql 21.72533, 21.q46~8 22.07h11 21.11110 2 1. 1114221.161lt4 21. Q 13Q4 27.hR6tl 29.Q663d31.47314 21.l15QQ 21.22102 81.Q1R30

lOl.QO'i67 116.1481~ 11.11110 21.11'>1.723.41?AR 61.dQti45 ~39.64711 1167. C1520

1204. 'tAb S7155

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KSTOP= 5021.111,10 21.11110 21.11110 21. 1111021.1111~ 21.11110 21.11110 21.11110~1.llll.O- 21.11110 21.1\110 21.1111021.11ilC 21.11110 21.11110 21. 1111021.'11t~ 21.t\ll(' 21.1111() 21.1111121.11261 21.11346 21.1111~ 21. 1111021.111AO 21.12273 21.211Sc) 21.2044421.1Q70Q ll.12'46 21.31C29 22.~4540~2.7qql' 21.C746(\ , 21.11tl~ 21.1121521.23471 22.6854'. 31.712c.l 3'i.47940'A.n22~q 21.b3qql JC. R1241 102.11~30

lZQ.27733 14q.J~11J 21.11111 21.12~O6l4.779tl 75.40~A8 q17.~02?5 1176.91')68

1426.611'11K~TOP: 60

21.1111C 'I.Jlllt') 21.11tl0 21.1111lj21.1\110 ?1.11110 21.1111f' 21.1111021.IIIIC 71.1111" 21.11110 21.1111C21.1111" '1.11110 21.11110 21.1111011.1111~ '1.11110 21.1111C li.11127ll.1152;1 '1.1174~ 21.11110 21.1111321.11311 21.14006 11. ~2 306 21.)1018·'I •~ c)h 84 71.\4124 ll.50Ae;1 23.2203424.('')754 ~4.4q8h4 21.11111) 21.1136321.~512;» 71.7424C, 36.42712 41.'l11781t1).6C;f)~4 i!2.01141 'l4.QQ8SQ 122.('QZJO

1') 6 • '+ Ii f. C) h lR'. ?O~'i' 21.ll1tq 21.1217lJ26.)t)lQ6 fJ~.64Sq1 IlOl.41S1Q 1571.354001613.742Q;>I( STOP: 70

?1.1111f' 11.11110 11.11110 21.1111C21.11110 21.11110 ll.Il11(\ 21.1111~21.11110 ?l.lll~C ll.11110 21.111.1021.1111C 11.11110 21.11IIC 21.1111Q21.11125 21.11110 21.11110 21.1116521.12025 ~1.12Itq1 21.11\10 21.1112121.ithB4 21 • 1 71 Jq 2t.4Cl136 21.47844ll.8C38h 21.11255 ;> 1. QOC I'. 24. '468221).5h'lRC 2h.14!>~6 21.11110 21.1161821.1)3741 25.r6754 41.60;>4 'j 49.05173~4.2?368 12.4Q01" 3<). 32Hl3 141.41qq4

18'.11061 214.S2771 21.111Jh 21.13t157l7.Qt;4A' 1"1.41)647 1 '- 16 • 2 ~J?q1 1151.40614182 A• 1R·) t; ~

KSTOP= 8021.1111~ 21.11110 21.iltlC 21.1111021.11110 -71.11110 2!.tttl0 21.\111021.11111) 21.11110 11.1111C 21.1111021.1tllC 21.11110 21.11110 21.1114C21.1111)6 21.11110 't 21. 1111 C 21.112432t.12tJ~f:! 11.t17t1~ 21.11110 '1.11\!>721.1112ti 21.22103 21.12177 21.1222122.?1407 21.22165 22.19286 25. 71 1)0217.46:>c;Q 28. Sqf) 30 21.11110 21.1201821.1~O3~ 2"'.6l25'3 41.12115 '56.7nI90,61 •. '36714 23. (\ 12C 'I 43.Q8574 160.0646820Q.084C't llt~.~R14q 21.t116J 21.153032Q.6tflSQ 113.71'i18 1321 • -, 1 ~5 7 1q 2 r; • 21 753

2013.~1594

156

-

KSTOP= 9021.11110 21.11110 21.11110 21.1111021.11110 21.11110 21.1111t' 21.1111021.1111C 21.11110 11.11110 21.1111021.11110 2l.tl110 21.ll11C 21.1118221.1121~ 21.11110 21.1\11(\ 21.1138521.1"'~6 21.15182 21.11110 21.11211~1.l'414 11.?qS'i~ 21.038SQ 22.05l1Q22.7t.J11CJ 21. 2q 231 22.68Q12 21.2JJa942Q.6CjQllt 31.2117Q 21.1111Q 21.ll~q822.0881C 28.40578 52.81621 61t.1130072.qIt82~ 2J.75C12 IttJ.7Q191 178.1)0034

114.11'91) '76.7Q661 ? 1. 11103 21 • 1 11 4]'l1.4Q44C; 125.~"6dl 1418.QQb58 208~. 3blC8

21QO.4448'KSTtlP= 100

21.1111~ 21.11110 21.11110 21.1111021.1111~ 21.11110 21.1111C 21.1111021.11110 ZI.1111~ 21.11110 21.1111021.11111' 21.1111' 21.11116 21.1\25321.1t~14 11.11110 21.11111 21.1161821.1bZC6 21.1R6d6 21.11110 21.1130121.150Ql ·21.3Q146 22.42A13 27.4160313.50"~3 21.1877Q 2'.281')1 29 ••)1j9A911.124~4 14.15257 21.11131 21.1339422.46103 'O.'S51~ ~Jl. 7AC) 43 7l.Q7tJ1'la?RZ'lO l4.C)161)7 .. t;". f" Ii 1'. "1 195.23741

2';8.7CJ76': 'JOb.61184 '1.11~'iq 21.lq4r~33.~17q5 136.qOO1~ \')10.1'141)3 2?4 It. 'j 3540

ll5Q. 16~'~t<srop= 110

21.\1110 21.11110 21.tlll~ 21.1111C21.11110 21.\111'\ ?1.lIIIC '21.1111'21.11110 2·1. 11110 21.11111) 21. title21.1111C 21.1111':' 11.11121 21.111bH21.t\47' 11.11\10 21.1111Q 21.t197ti21.18«131) 21.?2f:84 21.11110 2t.11415~1.17t;lq 21.')1104 ?2.9~ql? 2?QQ"Qs24.34~25 21.51'=57 . 23.q~ ~78 30.qq88~

34.-9242'» 17.~1t:5b 71.11151 21.144422l. ~q61q 32.'.SOQ4 64.1A642 81.30~59q2.e~'30 2;.36310 ~8.C)b}lC 211.B04Q'l282.')11S~ 33c;.hC1S~ 11.11316 21.~210731).1111'» t47.1''tbc) l'lq6.187~1 21'13.17041

1')22.10801I( Srnp:: 12~

21.11110 21.11110 21.11110 21.1111021.11111) ll.11111) 21.\\110 ll.ltllO~1.11110 ?l.tlttn 21.11110 21.1111021.11110 · ~1.1111C 21.11147 21.1154J21.1170Fl 21.1\110 21.11134 21. 12500?l.'~'i'jh 21.71Qb' 21.11110 21.1161021. 2086~ 21.6QQZ3 23.4SC09 21.6236325. 37 1·3 ~ 21.hh2'i4 24.771t3S 11. '1829611.72;lQ 40.R4~1c; 21. III 8? 21.15771l3.1q2ftA 14. f)lJb2q 10.~'208 aCJ.888S5

103.045132 2b. ~313~ 63.46~ql ~27.73q64. :l()5~~2881 '61.73Q7C; 11.11416 Z1• 2 c;~ 10

:\ 7. 10417 15·Q.12170 1677.b6211 2'i36.5166C2619.QqQ16

157

-

I( STOP= 13021.11110 11.11110 21.1111C. 21.tlllC21.11110 ?1.11110 21.1111f) 21.1111(,;21.1111~ 21. t 1 11 t' ?t.tltle 11.111\521.1111C , I. 11 110 21.11180 21.1118321.1204'; ?l.llllO ll.l\l'iq 21.1'4l2621.211Q7 ?1.'ttt88 21.11110 21.11SQ721.?~'7q 21.q042~ 14.0A028 .~4.1S07226.41~Cj' 11.84482 2~.644bl) lS.~qt')R940. lq6q I 44.')14bJl 21.11?21 21.1741223.94511 '6.Q1774 76.886qq 98.42715

11"4.14716 27.16481 6Q.l18CQ 24J.OBO~SJ27.~'i4?!i 1Ql.077,Q 21.11"5.')2 21.7.SQC539.01187 168.0'lblb 1154.17783 2674.70()20

2~12.2348h

'<STOP= litO21.11110 21.11110 21.11110 21.1111021.11110 11.1111(\ 71.11110 21.1111021.1111~ 21.11110 21.1111C 21.111242'1 • 1111 C 21.11110 21.ll?14 2 1. 121 J(j21.11506 21.11110 21.11195 21.142Clll.32Q92 21.43CZ4 21.1111C) 21.1225621. ~Cq 14 22. 1416 /.) 14. 7872 'i · 25.1790Q'l1.66~9n· l2.r5A04 26.'iQ473 17.6C1Sd44.01202 48.35711 21.1121'3 21.1Q386~4.5S013 3Q.~6'il" 82.Q2,4 77 106. lllJ155

123.44164 2B.3041' 73.1676(\ Z'i 7. 3b5q734q.5~78~ 411.1>5&;16 21.1171q 21 • 3]:) 254C.Q2146 111. h 7'0 'i la2d.19492 2~O8.21141

2~7q.qC~lC

KSTOP= 1SO21.1111" 21.11110 21.1111(\ 21.11110

.21.1\110 ll.ll1l·.... 21.1111C 11.1111021.1111r· '1.1Itle 2l.1111r 21.111362l.tllle 21.1111C 21.11308 21.1257021.111'6 'l.llllr:' 21.11749 21.154682 1.401) 13 21.5~10~ 11.11121 21.t212~21.~7q01 22.")(' ~, l'i.5671Q 26.10~lb29.C225A 2?3~'34 27.6145'> 40.0039t347.34637 52. '34C3' 21. 1134q 21.l11\825.10361 41.81122 88.~2572 115.46153

133.5C;\4"'4 2Q.'Q48l • 17.\l4Cl~ 212.134C3310.58'\74 441.~1465 21. 11 q06 21.376374?A4750 lRb.As:t4QQ IIJ9A. ZQ7hl 2q31.48657

'121.4C674. K Srnp= 1hO

21.1111" 21.1111t) 21.1.1110 21.1111(,2 1 • tit 10 21.11110 21.11110 21.1111021.11110 21.11110 21.11118 21.1115721.11110. 21.11110 :. 21.11414 21. 131 5021. ll-lC ? ?l.llllO · 21.11325 21.17t)1021.48,q~ 21.650')3 21.11\43 21.13'0121.4"3SQ 22.7525Q 26.41A17 27.1ZR3330.4850'; 22.'5714l 28.~q,..q3 41.4172650.77SQ4 56.1t1184 21.11441 21.24'.25, 2'i.QOlS? 44.J0168 C)4. q 75 '3.' 1l1.Q0411

143.6707A '''.52a8~ 82.64H06 2S5.Q1SQS3Ql.OC;SlQ 4 b8. 6"263 21.1~131 21.4214Q44. 7688~ I?S.1514Q lQbS.41f\SQ 3062. ~ 7q39

326 3.0Q/t?4158

... .

K5TOP= 17021.1111{\ 21.11110 21.11110 21.1111021.11111) 21.1111~ 21.1111C 21.1111CJ21.11110 11.11110 21.11127 21.1118&21.11111) 21.11111 21.11555 21.1188921.11t~q4 '1.1\110 71.11424 21.1 qt)5421.5e20C; 11.1SQ69 21.1116'; 21. t 403421.S6'AQ. 21.1143C 2".~3434 28.2411331.04149 '2.8R214 2Q.8350a; 't5.01C45~4.2qll? 60.h26Af) 21.11554 21.27,)?526.61Qal, 46.R2436 100.76115 1,? 2 7252151.f6CCh 31.10tJ4 1'1.281)11) ZQ9.lIiZQ3itlC.Q711Q 4q 3. 22632 21. 12'93 21.483601t6.68SQa 2(14.31416 202<). 811 Cjl 1181t.10~1931qq.263q2

KSTor= 18')21.11111) 21.1111" 21.1111C 21.1111021.11110 21.11110 21.1111':' 21.1111021.1111(\ 11.11110 21.1114(' 21.1112b11.1111'" ?1.11IZ? 21. 11143 21.1't~Ol21.16121 21.1111J 21.11~51 21.2146121.6q51q 21.q4q14 21.\11412 21.1491521.6A071 23.~14q4 2A.1IllO 29.4396113.7C117 23.2175'\ 31.0l25R 47.5Q'3557.R674h 64.887~1 ?1.116Ql 21.1103227.414~" 4q.36qlq I.OIl. ~~l ~1 14t). 554 32

163.54477 32.QC7Ci'3 ql.84Rll 312.165531t30.]~~16 S17.1IjC)C) 21.1lbQa 21.544t!248.601)10 212. 5182~ 2091.71828 ,\]0'.235351~12.114Cl

K STOP= 19021.1111~ ~1.11110 21.1111C 21.1111021.11110 21.1111t) 21.11111: 21. 1111021.11110 11.11111 21.1115q 21.1128121.11110 21.11111 21.11QSJ 21.1';11221.1160Q 11.11110 21.1,1726 21.2""\3321.fJ23q3 22.13014 '1.11229 21.15q6711.81414 2,.qlj"\h~ 2Q.'l4717 30.7184335.1t315c; 23.~8C'Rl '2.25'\78 50. 2112361.4Q43Ci ·6Q.20113 21.11~58 21.~4q5328.2?29q 51.QZ783 I lZ. 32~20 148.74089

17"4.'1447 '4.14131 qb.33546 l24.68lA444q.261~3 1)40.49561 21.11051 21.6110550.S1118 220.~6~Ol 2151.3Q7Q5 3418.1,9151

3662.0S24C1I( STOP= 200

21.1111(\ 21.11110 21.11110 21.1111021.11l10 11.11110 21.11110 21.1111C21.1111C 21.11121 21.11185 21.1134921.11110 21.11150 '1.ll~85 21.1724121.1Cj388 ~1.11116 :, 21. 11940 21.2711121.9b813 22.'33252 21.11276 21.171C421.q6641i 24.42C;26 )0.43532 32.07Z1131.24860 '3. Q7151 33.52144 52.~647965 .160 2~ 73.'iSQ20 '1.12051 21.393052 9. ~60q6 1)1t.4q339 117.QQC71 156.8258fi,

182.q61JJl 15.40022 100. 1'. ')q (. 336.6325246 1.6Q6S", Cjh~.29310 21.11451 21. &823452.42186 228.3023A 2209.911.19 3531.30835

3789.10132

159

I

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~AlER'Al ~U~BER. 1. ~U~PfR CF Tf~PfRATU~E CAPes- 1. ~'SS CENSITY. C.lc:en Cl.~CClllS RaTfe-

CYltNCEA 1.5 • 1.~ t""RLF~ "Else" ' .. FIf"'l ~'RESS

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~l~PER OF ElE~E~l~---------- 1~

~u~eER CF CIFF. ~~TERlllS--- I

~l~8ER ~F PRESSU~F caRC~---- 0

"tAL ACCElERATICN---------- c.o

A~GUlAR ~ElCCtlY------------ 0.0

REFfRfNce Te~PFR~Tu~e~------ C.2111C C'

~C~PE~ CF ~PFRC~ t~'Tt~N~---- 1<:.locrr: Cl

Tf~PEPATURf

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•

Figure No.

APPENDIX fj"

LIST OF FIGURES

Page

1

2

3

~..

5

6

·1

8

9

10

Spall Shape and Associated Stresses

Potential Spall1ng Conditions in ~oncrete

Thermal Stress Distribution from LaserRadiation

Nominal Uniform Compressive Test

Brittle Fracture Criterion in TwoDimensions

Fracture in Simple Compression

Stresses after Fracture Initiation DuringThermal Spalllng

Nominal Stress Distribution Near Tip ofFracture

Vertical Crack in Thermal Spal11ng

Failure of Surface

19

25

26

27

31

33

41

42

I11

12

13

Observed and Caclulated Temperature at 68Backside Centerspot of Disc

Comparison of Experimental and Analytical 70Values of Membrance St~a1ns

Stresses in Granite Cylinders as a Function 75of Specimen Length

14

,

Temperature and Stress Distribution for aGranite Cylinder" (50 Watts, O~4 SecondsHeating)

166

81

15 Maximum Surface Temperature Vs. Heating 82Time for .Granite Cylinders

16 Stress Vs. Heating Time for Granite 83Cylinders·

17 Variation with Respect to Heating Time of 8ijthe Ratio of Maximum Compressive Stress overthe Maximum Spall Producing Tensile Stressin Granite Cylinders

18 Thermal Energy Required to Cause Spalllng Vs. 85Power Level ·

Figure No.

APPENDIX D

LIST OF' FIGURES(Continued)

Page

19 ~emperature and Stress Distribution for a 89Marble Cylinder (50 Watts, 0-6 SecondsHeating)

20 Maximum Surface Temperature Vs. Heating Time 90for Marble Cylinders

21 Stress Vs. Heating Time for Marble Cylinders 91

22 Variation with Respect to Heating Time of the 92Ratio 9f Maximum Compressive Stress over theMaximum Spall Producing Tensile Stress inMarble Cylinders

25 Stress Vs. Heating Time for Porcelain Spheres 99

Maximum Surface Temperature Vs. Heating Time 98for Porcelain Spheres

23

24

Temperature and Stress Distribution for aPorcelain Sphere (100 Watts, 1-25 SecondsHeating)

97

1-- " ,

167

I

Figure No.

26

27

28

29·

APPENDIX D

LIST OF FIGURES(Continued)

Variation with Respect to Heating Time ofthe Ratio of Maximum Compressive Stre8Sover the Maximum Spall Producing TensileStress in Porcelain Spheres

Grid for Heat Transfer

Grid for Stress Analysis

Fine Grid for Finite Element Programs

168

Page

100

132

131

138

APPENDIX E

LIST OF TABLES

Table No.Page

1Material Properties

61

2Summary of Experiments and Stresses

14

for Variation in Specimen Lenghts

3 Granite Cylinders {l~" by l~") Heated 80

Iwith a Laser Beam 0.2 inches in Diameter

4(11"

1

Marble cylinders by 1"2") Heated 882

with a Laser Beam 0.2 inches in Wide

5 Porcelain Sphere~(1~" diameter) Heated 96

with a Laser Beam 0.2 inches Wide

169