Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

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Piezonuclear Transmutations in Brittle Rocks under Mechanical Loading 379

Fig. 12. (a) Locations of the largest iron mines in the world [24-27]. Iron ore reservoirs

(Magnetite and Hematite mines) are located in geographic areas with reduced seismic risks

and always far from fault lines. (b) The largest aluminum (bauxite) reservoirs are reported

together with the main Andesitic formations and most important subduction lines and plate

tectonic trenches [17]

Fig. 13. The estimated mass percentage concentrations of Si, Al, Fe and Ni in the Earth

crust during the last 4.5 Billion years (age of the planet Earth) [5,17-19,22, 28-34]

380 A. Carpinteri et al.

10 Conclusions

Neutron emission measurements were performed on Green Luserna Granite speci-

mens during mechanical tests. From these experiments, it can be clearly seen that

piezonuclear reactions giving rise to neutron emissions are possible in inert non-

radioactive solids under loading. In particular, during compression tests of specimens

with sufficiently large size, the neutron flux was found to be of about one order of

magnitude higher than the background level at the time of catastrophic failure. For

test specimens with more ductile behaviour, neutron emissions significantly higher

than the background were found. Neutron detection is also confirmed in compression

test under cyclic loading and during ultrasonic vibration.

Our conjecture, also confirmed by the Energy Dispersive X-ray Spectroscopy

(EDS) tests, is that piezonuclear reactions involving fission of iron into aluminum,

or into magnesium and silicon, should have occurred during compression on the

tested specimens.

This hypothesis seems to find surprising evidence and confirmation also at the

Earth crust scale from both geomechanical and geochemical points of view. In this

way, the piezonuclear reactions have been considered in order to interpret the most

significant geophysical and geological transformations, today still unexplained.

Finally, through experimental and theoretical studies of neutron emission and

piezonuclear fission reactions from brittle fracture, it will also be possible to ex-

plore new and fascinating application fields, such as short-term prediction and

monitoring of earthquakes, production of neutrons for medical use in cancer ther-

apy, disposal of radioactive waste, and even the hypothetical production of clean

nuclear energy.

Acknowledgements

The financial support provided by the Regione Piemonte (Italy) RE-FRESCOS

Project, is gratefully acknowledged.

Special thanks are due to R. Sandrone and A. Chiodoni of the Politecnico di

Torino for their kind collaboration in the EDS analysis. The authors wish to thank

also D. Madonna Ripa and A. Troia from the National Research Institute of Me-

trology – INRIM, for their indispensable assistance during the ultrasonic tests.

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Stress Triaxiality at Crack Tips Studied by

Caustics

Emmanuel E. Gdoutos

Office of Theoretical and Applied Mechanics of the Academy of Athens

School of Engineering, Democritus University of Thrace

GR-671 00 Xanthi, Greece

egdoutos@civil.duth.gr

Abstract. The optical method of caustics constitutes a powerful tool in the hands

of the experimentalist for the solution of fracture mechanics problems. According

to the method, the area in the vicinity of the crack tip is illuminated by a light

beam and the reflected or transmitted light rays form an envelope in space. When

this envelope is cut by a screen a highly illuminated curve, the so-called caustic, is

formed. By measuring characteristic dimensions of the caustic the stress intensity

factor is determined. The method has been based on the assumption that plane

stress conditions dominate in the vicinity of the crack tip. However, experimental

evidence has shown that the state of stress is not pure plane stress. It changes from

plane strain near the crack tip to plane stress at a critical distance away from the

tip through a three-dimensional region. In order the caustic to be generated from

the light rays reflected or transmitted through the specimen from the plane stress

region certain conditions among the dimensions of the optical arrangement, the

specimen properties and specimen thickness need to be satisfied. These conditions

are investigated in this work so that the method of caustics can safely be used for

determination of stress intensity factors in fracture mechanics problems.

1 Introduction

The optical method of caustics has extensively been used for the determination of

stress intensity factors in crack problems under static and dynamic conditions

[1-7]. According to this method, the area in the vicinity of the crack tip is illumi-

nated by a light beam and the reflected or transmitted light rays form an envelope

in space. When this envelope is cut by a screen a highly illuminated curve, the so-

called caustic, is formed. By measuring characteristic dimensions of the caustic

the stress intensity factor is determined. For linear elastic crack problems the caus-

tic is the image of the circumference of a circle surrounding the crack tip, called

the initial curve, on the reference screen, and is directly related to the state of af-

fairs along the initial curve. For the determination of stress intensity factors, the

384 E.E. Gdoutos

state of stress along the initial curve must be known. This state changes with the

distance from the crack tip [8, 9]. Very near to the tip the state of stress is plane

strain, while after a critical distance from the tip it becomes plane stress. In the

area between the two regions the state of stress is three-dimensional. An investiga-

tion of the three-dimensional effects and the limits of applicability of the method

of caustics in crack problems was performed in [10-15]. An extension of the me-

thod caustics in conjunction with the method of birefringent coatings for the study

of crack problems in opaque materials was performed in [16-20].

The condition that the initial curve of the caustic lays in the region where plane

stress conditions dominate imposes certain restrictions on the values of the applied

load, specimen dimensions and characteristic lengths of the optical arrangement.

In the present work, the limiting values of these quantities are defined when the

specimen is illuminated by a parallel, divergent or convergent light beam. It is

shown that for the correct application of the method certain conditions among the

above quantities need to be satisfied. Thus, the experimentalist can safely apply

the method of caustics for the determination of stress intensity factors in fracture

mechanics problems.

2 Stress-Optical Equations

The stress optical equations governing the variation of the optical path of a light

ray traversing a transparent specimen or reflected from the front or the rear face of

the specimen are derived. The cases of plane stress and plane strain are considered

separately.

2.1 Plane Stress

The variation of the optical path Δs

t1,2

of a light ray normally traversing a speci-

men along the directions of the principal stresses σ

or σ

is due to the variation of

the index of refraction and the thickness of the specimen, and is given by

()

dnnnds

Δ−+Δ=Δ

02,1

2,1

(1)

where d is the thickness of the specimen, n

1,2

is the index of refraction along the

directions of the principal stress σ

1,2

, n the index of refraction of the specimen un-

der no load, and n

is the index of refraction of the surrounding medium. Δ de-

notes the variation of the respective quantity.

According to Neumann-Maxwell stress-optical law for linear elastic behavior

of the material, the variation of the refractive index Δn

1,2

due to loading along the

direction of the principal stress σ

1,2

is given by

(

)

31,222,112,12,1

εεε

++=−=Δ bbnnn (2)

where b

and b

are the strain optical constants and ε

, ε

are the principal

strains.

Stress Triaxiality at Crack Tips Studied by Caustics 385

For generalized plane stress conditions (σ

= 0) introducing Hooke’s law into

eq. (2) we obtain

1,222,112,1

σσ

CCn +=Δ (3)

where

() ()

[]

2122211

bbb

Cbb

C +−=−=

νν

(4)

with ν representing Poisson’s ratio.

From eqs (1) to (4) it is obtained that

(

)

dbas

ttt 1,22,1

2,1

σσ

+=Δ (5)

where

() ()

0201

, nn

Cbnn

−−=−−=

(6)

Equation (5) can be put in the form

()()

[]

dcs

ttt 2121

2,1

σσξσσ

−±+=Δ (7)

where

baba

−

(8)

For optically isotropic materials for which the index of refraction does not change

along the principal stress directions (b

= b

= b,

= 0), eq. (7) takes the form

()

dcss

ttt 21

σσ

+=Δ=Δ (9)

where

()()

[]

nnb

ttt

−−−===

ννα

(10)

Consider now the case of a light ray normally incident on the specimen, passing

through it and reflected from its rear face. The variation of the optical path of the

light ray along the directions of the principal stresses σ

and σ

is given by

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

−+Δ=Δ d

nnds

2,1

(11)

386 E.E. Gdoutos

For generalized plane stress conditions (σ

= 0), the Neumann-Maxwell and

Hooke’s law can be applied to eq. (11) rendering

(

)

dbas

rrr 1,22,1

2,1

σσ

+=Δ (12)

where

⎟

⎠

⎞

⎜

⎝

⎛

−−=

⎟

⎠

⎞

⎜

⎝

⎛

−−=

νν

(13)

Equation (12) can be put in the form

()()

[]

dcs

rrr 2121

2,1

σσξσσ

−++=Δ (14)

where

baba

−

(15)

For optically isotropic materials (b

= b

= b,

= 0), eq. (14) takes the form

()

dcss

rrr 21

σσ

+=Δ=Δ (16)

where

()

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

−−−===

cba

rrr

νν

(17)

Finally, consider the case of a light ray normally incident on the specimen and

reflected from its front face. The variation of the optical path of the light ray along

the directions of the principal stresses σ

and σ

for generalized plane stress condi-

tions is due to the thickness variation only and is given by

()

dcss

rrr 21

σσ

+=Δ=Δ (18)

where

−= (19)

2.2 Plane Strain

The variation of the optical path of a light ray normally traversing a specimen

along the directions of the principal stresses σ

and σ

under plane strain condi-

tions (ε

= 0, σ

= ν(σ

+σ

)) is obtained from eqs (1), (2) and Hooke’ law, i.e.,

Stress Triaxiality at Crack Tips Studied by Caustics 387

(

)

dbas

ttt 1,22,1

2,1

σσ

+=Δ (20)

where

()

[]

ννν

+−−= (21a)

()

[]

ννν

−++−= (21b)

For a light ray normally incident on the specimen, and passing through its thick-

ness and reflected from the rear face of the specimen, the variation of the optical

path is given by

(

)

dbas

rrr 1,22,1

2,1

σσ

+=Δ (22)

where

trtr

bbaa 2,2 == (23)

Finally, the variation of the optical path of a light ray reflected from the front face

of the specimen under plane strain conditions is zero.

3 The Optical Method of Caustics

In the optical method of caustics a specimen is illuminated by a light beam (Fig. 1)

and the reflected or transmitted rays undergo a change of their optical path caused

by the variation of the thickness and refractive index of the specimen. At stress

gradients resulting at crack tips, the light rays generate a highly illuminated three-

dimensional surface in space. When this surface is intersected by a reference

screen, a bright curve, the so-called caustic curve, is formed. The dimensions of

the caustic are related to the state of stress near the crack tip. An optical arrange-

ment for the method of caustics is shown in Fig. 2. For the case of a mode-I

through-the-thickness crack the stress intensity factor K

exp

is given by [1]

2/3

2/5

exp

0934.0

cdmz

K =

(24)

where z

is the distance between the specimen and the viewing screen where the

caustic is formed, c is the stress optical constant of the specimen under conditions

of plane stress, d is the specimen thickness, m is the magnification factor of the

optical arrangement defined as the ratio of a length on the reference screen where

the caustic is formed divided by the corresponding length on the specimen and D

is the transverse diameter of the caustic at the crack tip (Fig. 3). The above equa-

tion is valid when the state of stress in the vicinity of the crack tip is plane stress.

388 E.E. Gdoutos

Fig. 1. Optical arrangement for divergent (a), convergent (b) and parallel (c) light

The magnification factor of the optical arrangement for divergent light is given by

(25)

where z

is the distance between the light source and the specimen.

For convergent light, where z

is the distance between the focus of the imping-

ing light beam and the specimen m is given by

−

±=

(26)

m = 1 for parallel light.

The caustic is created by the light rays reflected from the circumference of a

circle, the so-called initial curve, which surrounds the crack tip. The radius r

the initial curve is given by

5/2

5984.0

Kcdz

(27)