MODERN SITUATION IN PHOTOACOUSTICS IN APPLICATION TO THE
PROBLEM OF MECHANICAL STRESS MEASUREMENTS
K.L.Muratikov, A.L.Glazov
Ioffe Physical-Technical Institute of RAS, Polytekhnicheskaya 26, 194021, St.Petersburg, Russia.
E-mail: [email protected]
ABSTRACT
Experimental results obtained by the photoacoustic method for Vickers indentation zones in ceramics and metals are
presented. The influence of the given external stresses on the photoacoustic images of Vickers indentations is demonstrated
both for ceramics and metals. Obtained experimental results are compared with the available theoretical models of the
photoacoustic effect in stressed materials.
Introduction
The problem of residual stress detection and measurement is urgent in modern engineering, mechanics and material physics.
A good deal of effort has been undertaken in developing various methods for the residual stress detection [1]. Such methods
as Raman spectroscopy, optical, ultrasonic, magnetic, X-ray and neutron diffraction methods are already applied effectively for
this purpose in many cases. Hole drilling and compliance methods are actively developed at present for residual stress
detection. Recently holographic and speckle interferometry has attracted serious attention for solving the problem [2]. At
present thermoelastic stress analysis (TSA) is also a well-known experimental technique providing information on surface
stress fields in structures [3]. Most of these methods are limited in application. For instance, Raman spectroscopy is mainly
used in science and technology of semiconductors for which the phonon spectra have a relatively simple structure and line
shifts with stresses are well known. The TSA and holographic interferometry are of more general applicability, but have
comparatively low spatial resolution.
From this viewpoint photoacoustic (PA) methods are attractive for modern diagnostics and imaging of near surface structures.
The PA methods have micrometer resolution, are non-destructive and appeared to be useful for the detection of cracks and
voids, delaminations and possible second phase material inclusions in the near surface layer of components. The first PA
detection of residual stress took place about sixteen years ago [4]. Since then, there were a number of attempts for similar
applications of the PA effect [5-9]. Nevertheless, residual stress detection by the PA methods has not been widely accepted. In
our opinion, this situation is primarily due to the lack of in-depth systematic studies of the PA effect in solids with residual
stresses including direct confirmation of stress influence on the PA signal. Accounting this we have performed detailed
experimental and theoretical investigation of the PA effect in stressed solids.
The aim of the work is to present our crucial results instantiating modern situation in the field of the PA detection and imaging
of residual stresses. Some of the experimental results were presented and discussed in details elsewhere [10-13]. The paper
presents also new results demonstrating the influence of external stresses on the PA images of Vickers indented ceramics and
metals. We hope that presented results show the near-term outlook of some possible PA microscopy applications for residual
stress detection.
Theoretical Model of Photoacoustic Thermoelastic Effect in Stressed Solids
To consider the theoretical foundation of the PA approach to the problem of residual stress detection we use the modern
theory of thermoelasticity for elastic solids. By introducing the thermoelastic parameter instead of the coefficient of linear
expansion in the equation of motion, the latter may be represented as [14]
r
r
r
ρu&& = μ∇ 2u + (λ + μ )graddivu − (3 λ + 2 μ )ρCε K ε gradT ,
(1)
r
where u denotes deformations of the body, λ and µ are Lame’s elastic constants, ρ and Cε are the density and specific heat of
the body, respectively; K ε is the thermoelastic parameter of the material, T is the temperature of the body.
It is of general importance both for the PA technique that the thermoelastic parameter depends on stress [15]. By
thermodynamic methods it was shown in [15] that in adiabatic conditions the thermoelastic parameter is related to the first
stress invariant σ by the relation
1 ⎛
1 ∂E ⎞
K ε = K ε( 0 ) + K ε(1) =
(2)
σ⎟,
⎜ αT − 2
ρCε ⎝
E ∂T ⎠
where αT is the coefficient of linear thermal expansion, E is Young's modulus, σik are the components of the stress tensor,
α
σ = σ ii is the first stress invariant. K ε( 0 ) = T is the well-known relation for the thermoelastic parameter in the unstressed
ρC ε
state of the body; K ε(1) accounts the influence of the stress on the thermoelastic parameter in the explicit form.
A modulated laser radiation focused at the sample surface produces a cyclic local heat source Q. We suggest here that the
pump laser radiation is strongly absorbed by the sample and Q is the surface heat source. The heat source generates thermal
waves in the sample. The details of our PA microscopy setup for investigation of stressed materials are shown in Fig.1.
2κ
, where ω is the frequency of the
ω
pump beam modulation. According to Eq. (1) the periodic temperature gradient produces periodic local deformations in the
body. It is important that the term with the temperature gradient includes the thermoelastic parameter. It means that laserinduced deformations of the body depend on local stresses in the small vicinity of the laser spot. In the PA microscopy these
deformations are detected by a piezoelectric transducer attached to a sample under investigation. We have shown elsewhere
[16, 17] that the PA images display mainly the near-surface thermoelastic properties of the sample.
Thermal waves are strongly damped and their propagation is limited by the length lT =
To get an expression for the PA signal we need to know the strain of the sample in the place of the piezoelectric transducer
attachment. To obtain this strain Eq. (1) have to be solved. But in the general case it is a complicated mathematical problem.
Therefore, we have analyzed this problem within the framework of the perturbation theory in quasi-static approximation. This
approach assumes that elastic and thermoelastic inhomogeneties of the object are small and the length of acoustic waves
exceeds essentially the length of thermal waves in a material. The second assumption is true for the most of the PA
microscopy experiments. In this case the PA signal determined by thermoelastic inhomogeneties of a sample can be
represented in the form of the volume integral
LASER
ACOUSTO-OPTICAL
MODULATOR
EXTERNAL
PRESSURE
SAMPLE
PIEZOELECTRIC
DETECTOR
X-Y STAGE
LOCK-IN AMPLIFIER
WAVE GENERATOR
Figure 1. PA microscopy setup for imaging of stressed samples.
r
r r
r
V (r0 , ω ) = C ∫ K ε(1) (R )T (0 ) (R − r0 , ω )dv ,
(3)
(0)
where C is the coefficient depending on the piezoelectric sensor characteristics, T is the non-stationary distribution of the
r
r r
r
temperature generated by a pump laser radiation in the homogeneous sample, R = (x, y , z ) , r = R
= (x, y ,0 ) , r0 = (x 0 , y 0 ,0)
z =0
denotes a position of the center of the pump laser beam at the sample surface, z-axis is perpendicular to the sample surface.
Using Eq. (2) for the thermoelastic parameter and Eq. (3) one can relate the PA signal detected by a piezoelectric transducer
with the near surface mechanical stress
(
)
r
r
r r
r
V (r0 , ω ) = C ' ∫ σ xx (R ) + σ yy (R )T ( 0 ) (R − r0 , ω )dv ,
where C ' = −
(4)
1 1 ∂E
C , and σzz is assumed to be very small near the sample surface.
ρCε E 2 ∂T
Eq. (4) shows that the dependence of the PA signal on stress is similar to the dependence of the TSA signal that is also
proportional to σxx + σyy [18].
Application of Proposed Theoretical Model to Analysis of Photoacoustic Signal Behavior Near Vertical Crack Tips
Eq.(4) was obtained under rather general conditions and demonstrates the dependence of the PA signal on stress. This
equation can be applied for analysis of the PA signal from stressed objects in various cases. In particular, it can be used for
the study of the PA signal behavior near various defects where mechanical stresses are usually concentrated.
We have applied the obtained expression, for example, for analysis of the PA signal behavior near the radial crack tips in
ceramics with Vickers indentations [19]. As it was noted afore, only a thin near surface layer is responsible for the formation of
the PA signal. Therefore, the radial cracks can be considered as plane ones for the PA analysis. It is known that the
components of the stress tensor near the plane crack tip in a thick sample are determined by equations [20]
σ xx =
σ yy =
cos
K II
3θ ⎞
θ⎛
θ
3θ ⎞
θ⎛
θ
sin ⎜ 2 + cos cos
⎜ 1 − sin sin
⎟−
⎟,
2⎝
2
2 ⎠
2⎝
2
2 ⎠
2πr
(5.а)
cos
K II
θ⎛
θ
3θ ⎞
θ
θ
3θ
,
sin cos cos
⎜ 1 + sin sin
⎟+
2⎝
2
2 ⎠
2
2
2
2πr
(5.b)
KI
2πr
KI
2πr
where КI and КII are the stress intensity factors which define the normal and shear components of the total stress; r, θ are the
polar coordinates with the origin at the crack tip (see Fig 2.).
The stress intensity factors of a crack in a general case are determined by the total action of residual stress and stress
produced by external loading. In linear crack mechanics the total stress intensity factors of a crack can be represented in the
form
КI = КI
(0)
(0)
(1)
+ КI ,
(0)
КII = КII
(1)
+ КII ,
(6)
(0)
where КI and КII
are the stress intensity factors related to residual stresses, K I(1) and K II(1) are the stress intensity factors
characterizing the crack behavior under external loading.
Based on Eqs. (4-6) one can obtain the PA signal near the radial crack tip in the form
(
)
r r
r
r
V (r0 , ω ) = C ' ∫ σ (r )T (0 ) R − r0 , ω dv ,
where
r
σ (r ) =
r
r
θ (r )
θ (r )⎤
2 ⎡ (0 )
(1)
(0 )
(1)
K
K
K
K
+
cos
−
+
sin
I
II
II
πr ⎢⎣ I
2
2 ⎥⎦
(
)
(
)
If stresses do not vary essentially at distances compared with the thermal wavelength lT , Eq.(7) can be simplified
(7)
(
)
(
)
r
r
θ
θ ⎤
⎡
V (r0 , ω ) = C " f (r0 , ω )⎢ K I( 0 ) + K I(1) cos 0 − K II( 0 ) + K II(1) sin 0 ⎥ ,
2
2⎦
⎣
(8)
r r
r
T ( 0 ) (R − r0 , ω )
2
dv , r0 and θ0 are the polar coordinates of the laser spot center.
where C ′′ =
C ′ , f (r0 , ω ) = ∫
π
r
r r
r
r
It should be noted that at r0>>lT the function f (r0 , ω ) can be approximate by f (r0 , ω ) ≅ ∫ T ( 0 ) (R − r0 , ω )dv
r0 .
Let us consider more thoroughly the PA signal behavior near the radial crack tips of Vickers indentation under external loading.
The stress intensity factors of the radial cracks for residual stresses produced by Vickers indentation are given by expressions
[21]
KI(0) = χ
P
3
L
,
КII
(0)
= 0,
(9)
2
where P is the indentation loading, χ is the dimensionless coefficient determined by the shape of the crack, and L is the length
of the crack.
The stress intensity factors that take into account the external loading of a crack depend on both the value of this loading and
the angle φ between the crack and external stress direction (see Fig. 2). For a vertical crack these stress intensity factors are
determined by equations [21]
K I(1) = ψI σ app L sin2 φ ,
K II(1) = ψ II σ app L sin φ cos φ ,
(10)
where, ψI and ψII are the dimensionless coefficients determined by the crack shape, σapp is the stress produced by external
loading.
Using Eqs. (4-10) the PA signal near the radial crack tips can be expressed in the form
⎡⎛ P
⎞
r
r
θ
θ ⎤
V (r0 , ω ) = C " f (r0 , ω )⎢⎜ χ 3 + ψ I σ app L sin 2 φ ⎟ cos 0 − ψ II σ app L sin φ cos φ sin 0 ⎥ .
⎟
2
2⎥
⎢⎜⎝ L 2
⎠
⎣
⎦
(11)
Eqs. (7), (8) and (11) can be used for analysis of the PA signal behavior near the radial crack tips. In accordance with these
equations, both the residual and external stresses influence on the PA signal. Eq. (11) shows that, in principle, the PA
measurements can provide an estimation of the stress intensity factors of cracks. Another important result following from Eq.
(11) is that external normal and shear stresses contribute to the PA signal near radial crack tips in different ways. The
θ
contribution of normal stress in the PA signal is proportional to cos 0 , while the contribution of shear stresses is proportional
2
σapp
normal
stress
y
r
r
r
r0
θ θ0
shear stress
x
φ
Figure 2. The configuration of a crack and external loading.
θ0
. This difference can be used for a separate measurement of normal and shear stresses from the PA measurements
2
r
r
near the radial crack tips. It should be noted that the function f (r0 , ω ) depends on the temperature distribution T ( 0 ) (R, ω )
inside the homogeneous sample and is known as soon as the homogeneous thermal problem solved.
to sin
Experimental Results of Photoacoustic Study of Ceramics with Residual Stresses
Experimental investigations presented in this work were performed by the PA microscopy method on hot pressed silicon nitride
ceramic. The PA signal was detected by a piezoelectric transducer with a resonance frequency of about 140 kHz that was
attached to the rear side of a sample, as shown in Fig. 1. The pump laser radiation was focused on the front sample surface in
a spot with the diameter of about 1 μm. The depth of thermal wave penetration in silicon nitride ceramic was about 10 μm at
modulation frequency 140 kHz according to our estimation. The PA experiments were performed on ceramic samples with
3
dimensions 5x4x1.5 mm . The PA images were formed using the computerized setup that moves the sample along two
coordinate axes with 2.5 μm step.
The main attention in this work was paid to investigation of the influence of external and machining stresses on the PA images
of this ceramic. Residual stresses were introduced in ceramic samples by Vickers indentation. This is the most reliable and
reproducible method for generation of residual stress and crack systems in solids [22]. All presented indentations were made
at the load 98 N. Vickers indentation has a rather complicated structure. However, in this work the main attention was paid
only to investigation of the PA signal behavior near the radial crack tips where strong residual stresses are usually
concentrated.
The influence of external loading on the PA response was investigated in direct experiments with silicon nitride ceramic.
Compressive external loading was applied parallel to the sample surface. Typical PA piezoelectric images of the Vickers
indented area in silicon nitride ceramic in initial and loaded states are presented in Fig. 3. First of all, in Fig. 3 one can see a
well detectable increase of the PA signal near the tips of radial cracks 1 and 3. In our previous PA experiments with annealing
of ceramics [23, 24] it was shown that the increase corresponds to residual stress fields near the tips of radial cracks. It
gradually disappears with annealing time. Near radial cracks 2 and 4 there is no the PA signal increase. Therefore, one can
suppose that cracks 2 and 4 are initially in the unstressed state. It should be noted that we have observed the same signal
behavior for some other Vickers indentations in silicon nitride ceramics and for a number of indentations in Al2O3-SiC-TiC
composite ceramic earlier [25-27]. It was supposed in [6] that some cracks are invisible for certain modes of the piezoelectric
element operation. We have performed the PA imaging of ceramic samples without indentations and obtained approximately
2.48
1
σapp
2.0
2
1.5
4
3
1.0
100 μm
0.57
a
b
Figure 3. The PA piezoelectric image of Vickers indentation in Si3N4 ceramic sample. A – without external load, b –
under external compressive stress 95 MPa.
the uniform PA signal for the piezoelectric element operation mode. Therefore, the difference in the behavior of the PA signal
near the tips of radial cracks 1 and 3 and radial cracks 2 and 4 was not related only with the piezoelectric element operation.
Much more likely, it reflects the combined effects of the radial and subsurface lateral cracks and the residual stress field after
Vickers indentation.
In Fig. 3b one can see modifications of the PA signal near cracks 1 and 3 under external loading. For these cracks the angle
φ ≅ 96° , therefore the external loading produces almost normal compressive stress near their tips. This compressive stress
partially compensates initial tensile stresses near the tips of cracks 1 and 3, which reduces the PA signals in these areas. Fig.4
presents more comprehensively the PA signal for both initial and loaded states along a line perpendicular to crack 1 and
crossing its tip. The PA signal behavior for crack 3 is similar. For a comparison, the results of calculations based on Eqs. (10)
for the signal part related with the initial normal stress distribution are also presented in this figure. The experimental results
are in a reasonable agreement with our theoretical model at short distances from the crack tips. Some disagreement between
experimental and theoretical results at larger distances is due to an asymptotic character of the expressions used for stress
tensor components in Eqs. (5) and (6) which are valid only for short distances. Fig. 4 shows that the external compressive
stress compensates partially tensile residual stress acting near the tip of crack 1. It should be noted that an external stress
does not produce any noticeable effect on the PA signal near cracks 2 and 4. This fact is also in a good correspondence with
Eq.(9). For these cracks the angle φ is as small as 3º-5º. And Eq. (9) predicates that under this condition neither the normal
nor shear stresses influence essentially on the PA signal. The unstressed cracks like cracks 2 and 4 are very suitable for
demonstrating the external shear stress influence on the PA signal, because the contribution of the PA signal component
(0)
related with the stress intensity factor КI is absent. Unfortunately, for cracks 2 and 4 the angle φ appeared to be too small.
For this angle the influence of external shear stresses on the PA signal is very weak.
In the experiments with Al2O3-SiC-TiC composite ceramic we succeeded to observe the influence of external shear stresses
on the PA signal [25-27] for cracks with the angle φ ≅ 17° . Experimental results show that the compensation of residual
stresses near the tips of these cracks 1 takes place under the external compressive stress about 340 MPa. The radial crack
length in Al2O3-SiC-TiC composite generated by Vickers indentation at 98 N was equal to about 150 μm. The given data make
it possible to obtain a constraint on the coefficients χ and ψ. From our data one can conclude that χ ≅ 0.075 ψ . It should be
noted that this estimation is in a good quantitative coincidence with the result obtained in the paper [28] for ceramics with Al2O3
grains. One can also conclude from the obtained experimental data that K I(1) ≅ K II(1) . The last result is in a good agreement with
the theory of the straight cracks in thick plates [20] that gives the exact equality for these values. Principally the approach
based on the PA method is able to provide the values of stress intensity factors.
Thus, the results obtained for silicon nitride ceramic and our previous results for Al2O3-SiC-TiC composite ceramic
demonstrate the usefulness of the PA microscopy for imaging of stress fields near indented areas and cracks in ceramics.
Photoacoustic imaging of Vickers indented metals
Experimental part of this work devoted to the PA effect in stressed metals was performed on nanonickel and nanocopper
samples. The front surfaces of these samples were polished to optical quality with 1 μm diamond paste. Residual stresses in
the samples were produced by Vickers indentation. The indentation load was 98 N for all samples. Unlike ceramics, no cracks
were produced in these metals by Vickers indentations. The nanonickel and nanocopper samples were as large as
3
3
4.8x3.2x2.9 mm and 4.4x4.4x2.9 mm , respectively. In this work more detailed consideration is given to the PA signal
behavior inside Vickers indented zones in metals.
1.8
PA signal (rel. u.)
1.6
1.4
1.2
1.0
0.8
-100
-50
0
50
Distance, μm
100
Figure 4. The behavior of the PA piezoelectric signal across the tip of radial crack 1. Circles corresponds, to the sample
without external loading, crosses corresponds to the sample under the external compressive stress 95 MPa. A solid line is
a theoretical distribution of normal residual stress across the crack tip.
In Fig. 5 the PA piezoelectric images of Vickers indented nanonickel sample in initial and loaded states are presented. One
can see that strong image modifications take place with the application of external loading inside the indenter print, whereas
the PA signal variations outside this zone are much less pronounced. For example, the first image of the sample in initial state
is almost 4–order central symmetrical (Fig.5a), while the image for the uni-axially stressed sample is only 2–order central
symmetrical (Fig.5b).
1.9
σapp
1.5
1.0
200 μm
200 μm
0.57
a
b
Figure 5. The PA piezoelectric images of Vickers indentation in nanonickel. A – for the free sample, b – for the sample
under the external compressive stress of 30 MPa.
The changes take place first of all in the brightness of diagonals in indented zone under external loading. After canceling the
load of 30 MPa the sample has almost returned in the initial state.
It should be noted that some of the PA images of Vickers indented areas in nanonickel were not 4–order central symmetrical in
the initial state. We suppose that strong residual stresses were in these regions before the indentation, because the images in
these cases resemble the images of the Vickers indentations under external uni-axial loading like presented in Fig.5b.
For the 4–order central symmetrical PA images of indented areas we have also investigated the behavior of the PA signal
outside Vickers indented area. In accordance with the theoretical model developed in [29] the stress tensor outside the
indentation zone depends approximately inversely proportional to the square of the distance from the indentation center. Our
experimental data for the PA imaging of Vickers indentations in nanonickel were in reasonable agreement with this result [30].
However, it should be noted that this effect was considerably smaller than the PA signal changes taking place inside the
indentation area under loading.
200 μm
2.24
σapp
2.0
1.5
1.0
0.5
a
b
c
Figure 6. The PA images of Vickers indented area in nanocopper. A – for the sample without external loading, b – for
the sample under external compressive stress 80 MPa, and c – the image of the sample after canceling the loading.
Let us consider one more application of the PA microscopy to imaging of stressed metals. In Fig. 6 three PA images of the
Vickers indented area in nanocopper under different conditions are presented. The first of them is 4–order central symmetrical
image corresponds to the initially unloaded state of the sample (Fig. 6a). The second PA image is 2–order central symmetrical
one with one bright diagonal (Fig. 6b). This PA image corresponds to the sample in which 80 MPa compressive uni-axial
stress was produced by external loading. Serious modifications of the PA images of Vickers indented areas in nanocopper
produced by the external loading are clearly seen in Figs. 6a and 6b. After unloading the sample we obtained 2–order central
symmetrical PA image but with another bright diagonal (Fig. 6c). One can suppose that these strong modifications of the final
PA image in comparison with the initial one are due to appearance of new plastic deformations under loading and subsequent
unloading the nanocopper sample.
The obtained results demonstrate the influence of external stresses on the PA images of Vickers indented metals. However,
more detailed investigations should be performed to separate the effects produced by stresses and plastic deformations.
Conclusion
The theoretical model of the PA effect in stressed materials is presented. The results presented in this work demonstrate the
feasibility of the PA piezoelectric imaging for detection of residual stresses near the radial crack tips in ceramics. It is
experimentally shown that there is the influence of external mechanical stresses on the PA images of the radial crack tips. It is
demonstrated that our theoretical model of the PA effect in solids gives a good fit to experimentally found stress field variations
near the radial crack tips in ceramics. For metals it is shown that small modifications of the PA signal produced by residual
stresses are detectable outside Vickers indented areas. Essentially stronger influence of mechanical stresses on the PA
images was observed inside Vickers indented areas in metals. It is established that a strong loading of metal samples may
lead to irreversible modifications of the PA images of Vickers indented zones. It is pointed out that PA technique is able to
provide imaging of mechanical stress fields in different materials with the high spatial resolution.
Acknowledgments
This research was supported by RFBR under grant No.06-02-17148; some parts of this research were supported by CRDF
and by the Committee of Science and Education of St.Petersburg’s Administration.
References
1. Withers P. and Bhadeshia H., Residual stress. Parts I and II. Mater. Sci. Technology , 17 , 355-375, 2001.
2. Pechersky M., Determination of residual stresses by thermal relaxation and speckle correlation interferometry.
Strain , 38 , 141-149, 2002.
3. Pitarresi G. and Patterson E., A review of the general theory of thermoelastic stress analysis. J. Strain Analysis ,
38 , 405-417, 2003.
4. Cantrell J., Qian M., Ravichandran M. and Knowles K., Scanning electron acoustic microscopy of indentation induced cracks and residual stresses in ceramics, Appl. Phys. Lett., 57, 1870-1872, 1990.
5. Burbelo R., Gulyaev A., Robur L., Zhabitenko M., Atamanenko B. and Kryl Ya., Photoacoustic visualization of
residual stress in ceramic materials, J. de Phys. 4, C7 , 311-314, 1994.
6. Zhang H., Gissinger S., Weides G. and Netzelman U., Detection of surface damage in ceramic by photothermal
and photoacoustic techniques, J. de Phys. 4, C7 , 603-606, 1994.
7. Burbelo R. and Zhabitenko M., Photoacoustic effect in elastic stressed regions of solids, In: Progress in Natural
Science, edited by Shu-yi Zhang, Taylor & Francis, London and Washington, Suppl., 6, 720-723, 1996.
8. Muratikov K., Glazov A., Rose D., Dumar J. and Quay G., Photodeflection and photoacoustic mycroscopy of cracks and
residual stresses induced by Vickers indentation in silicon nitride ceramic, Tech. Phys. Lett., 23, 188-190, 1997.
9. Muratikov K., Glazov A., Rose D. and Dumar J., Investigation of the influence of residual stresses on the
thermophysical and thermoelastic properties of silicon nitride ceramic by photothermal and photoacoustic
methods, Tech. Phys. Lett., 24, 846-848, 1998.
10. Muratikov K., Glazov A., Rose D. and Dumar J., Photoacoustic effect in stressed elastic solids. J. Appl. Phys., 88, 29482955, 2000.
11. Muratikov K. and Glazov A., Theoretical and experimental study of photoacoustic and electron-acoustic effects in solids
with internal stresses, Tech. Phys., 45, 69-76, 2000.
12. Muratikov K., Glazov A., Rose D. and Dumar J., Photothermal and photoacoustic measurement of thermal and
thermoelastic properties of ceramics with residual stresses, High Temperatures-High Pressures, 33, 285-292, 2001.
13. Muratikov K., Glazov A., Rose D. and Dumar J., Photoacoustics of the stressed state in solids. Rev. Sci. Instrum., 74,
3531-3535, 2003.
14. Landau L. and Lifshitz E., Theory of elasticity. Pergamon, New York, 1986.
15. Wong A., Jones R. and Sparrow J., Thermoelastic constant or thermoelastic parameter? J. Phys. Chem. Solids, 48, 749753, 1987.
16. Muratikov K., Laser photoacoustic imaging of inhomogeneous objects. Tech. Phys. Lett., 30, 956-958, 2004.
17. Muratikov K., Imaging inhomogeneous objects with free boundaries by laser photoacoustic method. Tech. Phys. Lett., 31,
839-842, 2005.
18. Stenley P. and Dulieu-Smith J., The determination of crack-tip parameters from thermoelastic data. Experimental
Techniques, 20, 21-23, 1996.
19. Muratikov K., Photoacoustics of the stressed state in solids. Rev. Sci. Instrum., 74, 722-724, 2003.
20. Sedov L., Mechanics of Continuous Medium, Nauka, Moscow, Russia, 1970.
21. Smith S. and Scattergood R, Crack-shape effects for indentation fracture toughness measurements. J. Am. Ceram. Soc.,
75, 305-315, 1992.
22. Cook R. and Pharr G., Direct observation and analysis of indentation cracking in glass and ceramics. J. Am. Ceram. Soc.,
73, 787-817, 1990.
23. Muratikov K., Glazov A., Nikolaev V., Rose D. and Dumar J., The influence of annealing on the behavior of the
photoacoustic and photothermal signals from Al2O3-SiC-TiC ceramic with residual stresses, Tech. Phys. Lett., 27, 500503, 2001.
24. Muratikov K. and Glazov A., Application of lasers in photoacoustic and photothermal microscopy of solids with residual
stresses. Proc. SPIE, 4680, 167-176, 2002.
25. Muratikov K., Glazov A., Nikolaev V., Rose D. and Dumar J., The effect of mechanical loading on the photoacoustic
response from radial cracks in Vickers indented Al2O3-SiC-TiC Ceramics, Tech. Phys. Lett., 2002, 28, 377-381, 2002.
26. Muratikov K., Glazov A., Nikolaev V., Rose D. and Dumar J., Thermoelastic photoacoustic effect near tips of radial cracks
in ceramics under external loading. High Temperatures-High Pressures, 34, 585-590, 2002.
27. Muratikov K. and Glazov A., Laser photoacoustic microscopy of solids with residual stresses. Proc. SPIE, 5066, 78-83,
2003.
28. Braun L.M., Bennison S.J., and Lawn B.R., Objective evaluation of short-crack toughness curves using indentation flaws:
case study on alumina-based ceramics, J. Am. Ceram. Soc., 75, 3049-3057, 1992.
29. Yoffe E., Elastic stress field caused by indenting brittle materials. Phil. Mag. A , 46 , 617-628, 1982.
30. Muratikov K., Glazov A. and Nikolaev V., Photoacoustic thermoelastic effect near Vickers indentations in
nanocrystalline nickel. Tech. Phys. Lett., 31 , 685-687, 2005.