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Richard B. HetnarskiEditor
Encyclopedia ofThermal Stresses
With 3310 Figures and 371 Tables
EditorProfessor EmeritusRichard B. HetnarskiDepartment of Mechanical EngineeringRochester Institute of TechnologyRochester, NY, USA
and
Naples, FL, USA
ISBN 978-94-007-2738-0 ISBN 978-94-007-2739-7 (eBook)ISBN Bundle 978-94-007-2740-3 (print and electronic bundle)DOI 10.1007/978-94-007-2739-7Springer Dordrecht Heidelberg New York London
Library of Congress Control Number: 2013951772
# Springer Science+Business Media Dordrecht 2014This work is subject to copyright. All rights are reserved by the Publisher, whether the whole orpart of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way,and transmission or information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed. Exempted from thislegal reservation are brief excerpts in connection with reviews or scholarly analysis or materialsupplied specifically for the purpose of being entered and executed on a computer system, forexclusive use by the purchaser of the work. Duplication of this publication or parts thereof ispermitted only under the provisions of the Copyright Law of the Publisher’s location, in itscurrent version, and permission for use must always be obtained from Springer. Permissions foruse may be obtained through RightsLink at the Copyright Clearance Center. Violations are liableto prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names areexempt from the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legalresponsibility for any errors or omissions that may be made. The publisher makes no warranty,express or implied, with respect to the material contained herein.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
Wear Rate of CCBC After ThermalTreatment
To evaluate the effect of crack induced by
temperature changes, thermal shock experiments
were followed by erosive wear tests. Results are
presented in Fig. 6. It was found that first thermal
cycle (cooling in air or water) have negative effect
on wear resistance of cermets. Four consequent
cycles 1,200 �C with air cooling decrease that
negative effect of the first cycle. For instance, the
erosion rate after 5-th cycle 1,200 �C with air
cooling is even decreased compared to as-received
samples. This phenomenon may be attributed to
redistribution of internal stresses induced during
sintering. Erosion rate of cermetwith 40wt%ofNi
shows that it is less affected by temperature
changes than cermet with 10 and 20 wt%.
Single Partial Immersion of CCBC
Partial immersion was found to be the most
effective way of differentiation of the thermal
shock resistance of CCBC having various metal
binder content.
Rectangular CCBC samples of 20 mm �12 mm � 5 mm size were polished from both
20 mm� 12 mm sides, heated with heating speed
of 400 �C min�1 up to 1,200 �C and then
immersed down to the depth of 1 mm into water
of room temperature.
It is possible to see (Fig. 7) that the CCBC
with 40 wt% of Ni has only one thermal crack
while cermets with low metal binder content
experience multiple cracking showing their
lower resistance to thermal shock.
References
1. Tinklepaugh JR (1960) Cermets. Reinhold, New York
2. Antonov M, Hussainova I (2010) Cermets surface
transformation under erosive and abrasive wear. Tribol
Int 43:1566–1575
3. Thuvander M, Andren HO (2000) APFIM studies of
grain boundaries: a review. Mater Char 44:87–100
4. Pierson H (1996) Handbook of refractory carbides and
nitrides: properties, characteristics, processing, and
applications. Noyes, New Jersey
5. Kaye G, Laby T (1995) Tables of physical and
chemical constants, 16th edn. Longman, London
6. Santhanam AT, Tierney P, Hunt JL (1990) Cemented
carbides. In: ASM handbook, vol 2 (Properties and
selection: nonferrous alloys and special-purpose
materials). ASM International, New Jersey
7. Lanin A, Fedik I (2008) Thermal stress resistance of
materials. Springer, New York
Further ReadingAntonov M, Hussainova I (2006) Thermophysical
properties and thermal shock resistance of chromium
carbide based cermets. Proc Estonian Acad Sci Eng
12(4):358–367
Pirso J, Valdma L, Masing J (1975) Thermal shock
damage resistance of cemented chromium carbide
alloys. Proc of Tallinn Tech Univ 381:39–45,
(In Russian)
Thermal Shock Resistance ofFunctionally Graded Materials
Zhihe Jin1 and Romesh C. Batra2
1Department of Mechanical Engineering,
University of Maine, Orono, ME, USA2Department of Engineering Science and
Mechanics, Virginia Polytechnic Institute and
State University, Blacksburg, VA, USA
Overview
This entry introduces concepts of the critical
thermal shock and the thermal shock residual
strength for characterizing functionally graded
materials (FGMs). It starts with the introduction
of basic heat conduction and thermoelasticity
equations for FGMs. A fracture mechanics-
based formulation is then described for comput-
ing the critical thermal shock for a ceramic-metal
FGM strip with an edge crack subjected to
quenching on the cracked surface. The through-
the-width variation of the shear modulus of the
FGM is assumed to be hyperbolic and that of
the thermal conductivity and the coefficient
of thermal expansion exponential. Finally a
ceramic-ceramic FGM strip with periodically
spaced surface cracks subjected to quenching is
Thermal Shock Resistance of Functionally Graded Materials 5135 T
T
considered to illustrate effects of material grada-
tion and surface crack spacing on the critical ther-
mal shock and the thermal shock residual strength.
Introduction
Functionally graded materials (FGMs) for high-
temperature applications are macroscopically
inhomogeneous composites usually made from
ceramics and metals. The ceramic phase in an
FGM acts as a thermal barrier and protects the
metal from corrosion and oxidation, and the
metal phase toughens and strengthens the FGM.
High-temperature ceramic-ceramic FGMs have
also been developed for cutting tools and other
applications. The compositions and the volume
fractions of constituents in an FGM are varied
gradually, giving a nonuniform microstructure
with continuously graded macroscopic proper-
ties. The knowledge of thermal shock resistance
of ceramic-ceramic and ceramic-metal FGMs is
critical to their high-temperature applications. In
general, thermal shock resistance of FGMs can be
characterized by the critical thermal shock and
thermal shock residual strength. The critical ther-
mal shock describes the crack initiation resis-
tance of the material and may be determined by
equating the fracture toughness to the peak
thermal stress intensity factor at the tip of
a preexisting crack emanating from the surface
and going into the FGM body. The thermal shock
residual strength is a damage tolerance property
describing the load carrying capacity of a struc-
ture damaged with a thermal shock.
The residual strength method to find the ther-
mal fracture resistance of monolithic ceramics
was developed by Hasselman [1]. Micro-cracks
inherently exist in ceramics. When a ceramic
specimen is subjected to sufficiently severe ther-
mal shocks, some of the preexisting micro-cracks
will grow to form macro-cracks. Crack propaga-
tion in thermally shocked ceramics may be
arrested depending on the severity of the thermal
shock, thermal stress field characteristics, and
material properties. The measured strength of
a thermally shocked ceramic specimen generally
exhibits two kinds of behavior as shown in Fig. 1.
In the first case, the strength remains unchanged
when the thermal shock DT is less than a critical
value, DTc, called the critical thermal shock. At
DT ¼ DTc, the strength sR suffers a precipitous
drop and then decreases gradually with an
increase in the severity of thermal shock. In the
second case, the strength also remains constant
for DT < DTc; however, the strength does not
drop suddenly at DT ¼ DTc but decreases gradu-
ally with an increase in DT. The residual strengthmethod has been further developed to investigate
thermal shock behavior of monolithic ceramics in
the context of thermo-fracture mechanics (see,
e.g., [2–5]).
The critical thermal shock and residual
strength methods have been employed to
evaluate thermal shock resistance of ceramic
composites in recent years. Examples include
experimental investigations on fiber-reinforced
ceramic matrix composites [6], metal
particulate-reinforced ceramic matrix composites
[7], and ceramic-ceramic FGMs [8]. These exper-
imental studies showed that the residual strength
method is an effective and convenient approach
for evaluating thermal shock resistance of
ceramic composites. Jin and Batra [9], Jin and
Luo [10], and Jin and Feng [11] developed theo-
retical thermo-fracture mechanics models to
evaluate the critical thermal shock and thermal
shock residual strength of FGMs.
This entry introduces concepts of the critical
thermal shock and the thermal shock residual
DTc DT
sR
DTc
sR
DT
a b
Thermal Shock Resistance of Functionally GradedMaterials, Fig. 1 Thermal shock residual strength
behavior of ceramics; (a) thermal shock resistance drops
precipitously at DT ¼ DTc ; (b) thermal shock resistance
decreases gradually for DT > DTc
T 5136 Thermal Shock Resistance of Functionally Graded Materials
strength for characterizing thermal shock resis-
tance of FGMs. The basic thermoelasticity equa-
tions of FGMs are described in section
“Thermoelasticity Equations of FGMs.”
Section “An FGM Plate with an Edge Crack
Subjected to a Thermal Shock” considers
a ceramic-metal FGM strip with an edge crack
subjected to quenching on the cracked surface
and derives a fracture mechanics-based formula-
tion to determine the critical thermal shock.
Section “An FGM Plate with Parallel Edge
Cracks Subjected to a Thermal Shock” considers
a ceramic-ceramic FGM strip with periodically
spaced surface cracks subjected to quenching.
The effects of material gradation and crack den-
sity on the critical thermal shock and the thermal
shock residual strength are also examined for an
Al2O3/Si3N4 FGM plate in section “An FGM
Plate with Parallel Edge Cracks Subjected to
a Thermal Shock.”
Thermoelasticity Equations of FGMs
Thermal shock behavior of FGMs is generally
investigated in the standard micromechanics/
continuum framework, i.e., FGMs are treated as
nonhomogeneous materials with spatially vary-
ing thermomechanical properties that are found
by using the conventional micromechanics
models for homogenizing material properties of
composites. Moreover, an uncoupled approach is
adopted in which the influence of deformation on
temperature is ignored, and hence the tempera-
ture field is obtained independently of deforma-
tions. The heat conduction equation for the
temperature without consideration of a heat
source/sink is
@
@xikðxÞ @T
@xi
¼ rðxÞcðxÞ @T
@tð1Þ
where T is the temperature, t time, k(x) the space-
dependent thermal conductivity, r(x) the mass
density, and c(x) the specific heat. The Latin
indices have the range 1, 2, and 3, and repeated
indices imply summation over the range of the
index. Equation 1 has been written in rectangular
Cartesian coordinates (x1, x2, x3) which we will
sometimes also denote by (x, y, z).The basic equations of thermoelasticity
include the equations of equilibrium in the
absence of body forces
sij;j ¼ 0 ð2Þ
the strain–displacement relations for infinitesi-
mal deformations
eij ¼ 1
2ui;j þ uj;iffi � ð3Þ
the constitutive relation
eij ¼ 1þ nðxÞEðxÞ sij � nðxÞ
EðxÞ skkdij þ aðxÞ T � T0ð Þ
ð4Þ
and boundary conditions. In (2)–(4), sij denotestresses, eij strains, ui displacements, dij the
Kronecker delta, E(x) Young’s modulus, n(x)Poisson’s ratio, and a(x) the coefficient of ther-
mal expansion, and the FGM has been assumed to
be isotropic. A comma followed by index j
implies partial derivative with respect to xj.
Under plane stress conditions, the equilibrium
equations can be satisfied by expressing stresses
in terms of the Airy stress function F as follows:
sxx ¼ @2F
@y2; syy ¼ @2F
@x2; sxy ¼ � @2F
@x@yð5Þ
Use of the constitutive relation (4) and the
strain compatibility conditions derived from (3)
yields the following governing equation for the
Airy stress function for general nonhomogeneous
materials:
H2 1
EH2F
� �� @2
@y21þnE
� �@2F
@x2� @2
@x21þnE
� �@2F
@y2
þ2@2
@x@y
1þnE
� �@2F
@x@y¼�H2 a T�T0ð Þ½ �
ð6Þ
Thermal Shock Resistance of Functionally Graded Materials 5137 T
T
where H2 is the Laplace operator in the xy
plane. For plane strain deformations, E, n, and aare replaced by E=ð1� n2Þ, n=ð1� nÞ, and
ð1þ nÞa, respectively.In analysis of deformations of FGMs, E and n
and other material parameters are assumed to be
continuously differentiable functions of spatial
coordinates. They can be calculated from
a micromechanics model or can be assumed to
be given by elementary functions (e.g., power
law, exponential relation) which are consistent
with the micromechanics analyses.
An FGM Plate with an Edge CrackSubjected to a Thermal Shock
This section describes a fracture mechanics for-
mulation to calculate the critical thermal shock
for a ceramic-metal FGM strip of width b with an
edge crack subjected to quenching on the cracked
surface [9]; e.g., see Fig. 2.
Basic Equations
The through-the-width variation of the shear
modulus is assumed to be hyperbolic and that of
the thermal conductivity and the coefficient of
thermal expansion exponential. That is,
m ¼ m01þ bðx=bÞ
1� n ¼ 1� n0ð Þ egðx=bÞ
1þ bðx=bÞð7Þ
a ¼ a0eeðx=bÞ; k ¼ k0edðx=bÞ; k ¼ k0 ð8Þ
where m is the shear modulus, k the thermal
conductivity, and b, g, e, and d are material con-
stants given by
b ¼ m0m1
� 1; g ¼ lnð1þ bÞ þ ln1� n11� n0
;
e ¼ lna1a0
; d ¼ lnk1k0
ð9Þ
in which subscripts 0 and 1 stand for values of
the parameter at x ¼ 0 and x ¼ b, respectively.
The assumed Poisson’s ratio in (7) is subjected to
the constraint 0 n < 0.5, and the thermal diffu-
sivity is assumed to be constant for mathematical
convenience. This can be achieved by suitably
varying the specific heat.
With (7)–(9) and the assumption that a plane
strain state of deformation prevails in the body,
(6) and (1) can be written as
1� n20E0
H2 egðx=bÞH2Fh i
þ H2 ð1þ nÞaT½ � ¼ 0
ð10Þ
H2T þ db
@T
@x¼ 1
k0
@T
@tð11Þ
Temperature, Thermal Stress, and Thermal
Stress Intensity Factor
Assume that the FGM strip is initially at
a uniform temperature T0, the surface x ¼ 0 is
suddenly cooled to temperature Ta with the sur-
face x ¼ b kept at temperature T0. The tempera-
ture distribution in the strip obtained by solving
(11) is given by [12]:
b
T0Ta T0
y
x
Thermal Shock Resistance of Functionally GradedMaterials, Fig. 2 An FGM plate with an edge crack
subjected to a thermal shock
T 5138 Thermal Shock Resistance of Functionally Graded Materials
T�T0
DT¼e�d� e�dx�
1� e�d
þX1n¼1
Bne�dx�=2 sinðnpx�Þe�ðn2p2þd2=4Þt
ð12aÞ
where x* ¼ x/b, DT ¼ T0 – Ta, t ¼ tk0/b2 is thenondimensional time, and
Bn¼ 2np1�e�d
1�ð�1Þne�3d=2
ð3d=2Þ2þn2p2�e�d�ð�1Þne�3d=2
ðd=2Þ2þn2p2
" #;
n ¼ 1;2; ::::
ð12bÞThe above heat conduction problem repre-
sents an idealized thermal shock loading case,
i.e., the heat transfer coefficients on the surfaces
of the FGM plate are infinitely large which cor-
respond to the severest thermal stress induced in
the plate. In other words, the critical thermal
shock predicted by the current model would be
lower than that obtained using a finite heat trans-
fer coefficient. For the one-dimensional temper-
ature field T ¼ T(x, t) given in (12), the thermal
stress in the strip is given by
sTyy ¼� Eayðx�; tÞ1� n
þ E
1� n2ð ÞA0
�"b A22 � x�bA12ð Þ
ð10
Eayðx�; tÞ1� n
dx�
� b2 A12 � x�bA11ð Þð10
Eayðx�; tÞ1� n
x�dx�#
ð13aÞwhere yðx�; tÞ ¼ Tðx�; tÞ � T0; and constants
A11, A12, A22, and A0 are defined by
A11 ¼ðb0
E
1� n2dx; A12 ¼ A21 ¼
ðb0
E
1� n2xdx;
A22 ¼ðb0
E
1� n2x2dx; A0 ¼ A11A22 � A12A21
ð13bÞ
Now consider an edge crack of length a0in the FGM strip as shown in Fig. 2. The
integral equation for the cracked FGM strip is
given by [12]
ð1�1
1
s� rþ kðr; sÞ
’ðsÞe�ða0=bÞ½ð1þsÞ=2�gds
¼ � 2p 1� n20ffi �E0
sTyyðr; tÞ; jrj 1 ð14Þ
where
’ðxÞ ¼ @vðx; 0Þ@x
ð15Þ
with v(x, 0) being the displacement in the
y-direction at the crack surface, and k(r, s) is a
known kernel. According to the singular equation
theory [13], (14) has a solution of the form
’ðrÞ ¼ eða0=bÞ½ð1þrÞ=2�g cðrÞffiffiffiffiffiffiffiffiffiffiffi1� r
p ð16Þ
where cðrÞ is continuous on [�1, 1]. Normaliz-
ing cðrÞ by ð1þ n0Þa0DT, the normalized ther-
mal stress intensity factor (TSIF), K�I , at the crack
tip is obtained as
K�I ¼ ð1� n0ÞKI
E0a0DTffiffiffiffiffiffipb
p ¼ � 1
2
ffiffiffia
b
rcð1Þ ð17Þ
The value of the TSIF can be computed once
(14) has been solved.
Critical Thermal Shock
The TSIF in (17) is a function of time. The critical
thermal shock may be obtained by equating the
peak TSIF to the intrinsic fracture toughness
Kcða0Þ. The peak TSIF obtained from (17) is
KpeakI ¼E0a0DT
1� n0
ffiffiffiffiffiffiffipa0
pMax t>0f g �cð1;tÞ=2f g ð18Þ
Hence, the critical thermal shock is given by
DTc ¼ 1� n0ð ÞKcða0ÞE0a0
ffiffiffiffiffiffiffipa0
pMax t>0f g �cð1; tÞ=2f g ð19Þ
Thermal Shock Resistance of Functionally Graded Materials 5139 T
T
The corresponding critical thermal shock for
the cracked ceramic specimen is
DT0c ¼ 1� n0ð ÞKceram
Ic
E0a0ffiffiffiffiffiffiffipa0
pMax t>0f g �cceramð1; tÞ=2f g
ð20Þ
where cceramð1; tÞ is the solution for the
corresponding crack problem of a ceramic strip
and KceramIc is the fracture toughness of the
ceramic. It follows from (19) and (20) that [9]
DTc
DT0c
¼ 1� Vm a0ð Þ½ � 1� n201� n2ða0Þ
Eða0ÞE0
� �1=2
Max t>0f g �cceramð1; tÞ=2f gMax t>0f g �cð1; tÞ=2f g
ð21Þ
where the following intrinsic fracture toughness
model [14]
KcðaÞ¼ 1�VmðaÞ½ � 1� n201� n2ðaÞ
EðaÞE0
� �1=2
KceramIc
ð22Þhas been adopted, and Vm equals the volume frac-
tion of the metal phase which is determined from
the three-phase micromechanics model [15] and
the assumed shear modulus in (7) with Vm¼ 0 and
1 at x ¼ 0 and b, respectively. Figure 3 shows the
normalized critical thermal shock DTc=DT0c ver-
sus the nondimensional initial crack length a0/b fora hypothetical FGMwith (b, d, e)¼ (1, 1, 0) [9]. It
is evident that DTc for the FGM is significantly
higher than that for the ceramic. Hence, the
cracked FGM strip can withstand a more severe
thermal shock than the corresponding ceramic
strip without the crack propagating into the strip.
An FGM Plate with Parallel Edge CracksSubjected to a Thermal Shock
This section describes a fracture mechanics for-
mulation to calculate the critical thermal shock
and the residual strength for a ceramic-ceramic
FGM strip with an infinite array of periodic edge
cracks subjected to quenching at the cracked sur-
face. It also presents numerical results to illus-
trate effects of the material gradation profile and
the surface crack density on the thermal shock
resistance of the FGM strip [11]. The FGM is
assumed to have constant Young’s modulus and
Poisson’s ratio but arbitrarily graded thermal
properties along the width. While these assump-
tions limit the application of the model, there
exist FGM systems, e.g., TiC/SiC, MoSi2/
Al2O3, and Al2O3/Si3N4, for which Young’s
modulus is nearly a constant.
Temperature and Thermal Stress Fields
We consider an infinitely long ceramic-ceramic
FGM strip of width b with an infinite array of
periodic edge cracks of length a and spacing
between cracks H ¼ 2 h as shown in Fig. 4. The
thermal parameters of the FGM are arbitrarily
graded in the width (x-) direction. The strip is
initially at a constant temperature T0, and its
surfaces x ¼ 0 and x ¼ b are suddenly cooled to
temperatures Ta and Tb, respectively. Since the
bounding surfaces x ¼ 0 and x ¼ b are kept at
uniform temperatures, and material gradation and
cracking are in the x-direction, it is reasonable
to assume that the heat for short times flows in the
x-direction.
Normalized crack length , a0/b
No
rmal
ized
cri
tica
l tem
per
atu
re d
rop
0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
Thermal Shock Resistance of Functionally GradedMaterials, Fig. 3 Normalized critical thermal shock ver-
sus nondimensional initial crack length (After Jin and
Batra [9])
T 5140 Thermal Shock Resistance of Functionally Graded Materials
Jin [16] obtained the following closed-form,
short-time asymptotic solution of the temperature
field in the strip using the Laplace transform and
its asymptotic properties:
Tðx; tÞ � T0
T0 � Ta¼� rð0Þcð0Þkð0Þ
rðxÞcðxÞkðxÞ 1=4
erfc1
2bffiffiffit
pðx0
ffiffiffiffiffiffiffiffiffikð0ÞkðxÞ
sdx
0@ 1A� T0 � Tb
T0 � Ta
� �rðbÞcðbÞkðbÞrðxÞcðxÞkðxÞ 1=4
erfc1
2bffiffiffit
pðbx
ffiffiffiffiffiffiffiffiffikð0ÞkðxÞ
sdx
0@ 1A ð23Þ
where kðxÞ ¼ kðxÞ=rðxÞcðxÞ is the thermal diffu-
sivity, t ¼ kð0Þt=b2 is the nondimensional time,
and erfc( ) is the complementary error function.
The asymptotic solution given by (23) holds for
an FGM plate with continuous and piecewise
differentiable thermal parameters. The signifi-
cance of the solution lies in the fact that the
thermal stress and the thermal stress intensity
factor (TSIF) in the FGM plate induced by the
thermal shock reach their peak values in a very
short time. Thus, (23) may be used to evaluate
peak values of the thermal stress and the TSIF
which govern the failure of the material. The
thermal stress that causes edge cracks to propa-
gate is still given by (13) with the temperature
given by (23).
Thermal Stress Intensity Factor
The boundary and periodic conditions for the
crack problem shown in Fig. 4 are
sxx ¼ sxy ¼ 0; x ¼ 0;�1 < y < 1 ð24Þ
sxx ¼ sxy ¼ 0; x ¼ b;�1 < y < 1 ð25Þ
sxy¼0; 0<x<b; y¼nh; n¼0; 1;...; 1;
v¼0; 0<x<b; y¼ð2nþ1Þh; n¼0; 1;...; 1;
v¼0; a<x<b; y¼2nh; n¼0; 1;...; 1ð26Þ
syy ¼ �sTyy; 0 < x < a; y ¼ 2nh;
n ¼ 0; 1; . . . ; 1 ð27Þ
where sTyy is given by (13) with the temperature
given by (23), and v equals the displacement
of a point in the y-direction. Using the Fourier
transform/superposition approach, the above-
described thermal crack problem can be reduced
to finding a solution of the following singular
integral equation:
ð1�1
1
s�rþKðr;sÞ
’ðsÞds¼�2pð1�n2Þ
EsTyyðr;tÞ;
jrj1
ð28Þwhere the basic unknown is still defined in (15),
nondimensional coordinates r and s are defined as
b
H=2h
T0Ta Tb
y
x
Thermal Shock Resistance of Functionally GradedMaterials, Fig. 4 An FGM plate with an array of peri-
odic edge cracks subjected to a thermal shock
Thermal Shock Resistance of Functionally Graded Materials 5141 T
T
r ¼ 2x=a� 1; s ¼ 2x0=a� 1 ð29Þ
and K(r, s) is a known kernel [11].
According to the singular integral equation
theory [13], the solution of (28) has the
following form:
’ðrÞ ¼ cðrÞffiffiffiffiffiffiffiffiffiffiffi1� r
p ð30Þ
wherecðrÞ is a continuous and bounded function.Once (28) has been solved, the TSIF at a crack tip
can be computed from
K�I ¼ ð1� nÞKI
Ea0DTffiffiffiffiffiffipb
p ¼ � 1
2
ffiffiffia
b
rcð1Þ ð31Þ
where KI denotes the TSIF, K�I the
nondimensional TSIF, DT ¼ T0 � Ta, and
a0 ¼ að0Þ. In (31), cð1Þ is a function of
nondimensional time t, the nondimensional
crack length a/b, the crack spacing
parameter H/b, and the material gradation
parameter.
Critical Thermal Shock and Residual Strength
As stated in section “Critical Thermal Shock”,
the critical thermal shock DTc that causes the
initiation of the parallel cracks of length a may
be obtained by equating the peak TSIF to the
fracture toughness of the FGM, i.e.,
Max t>0f g KIðt; a;DTcf g ¼ KIcða0Þ ð32Þ
whereKIc(a) is the fracture toughness of the FGM
at x ¼ a. Substitution from (31) into (32) yields
the critical thermal shock:
DTc¼ ð1� nÞKIcða0ÞEa0
ffiffiffiffiffiffipb
p Max t>0f g � 1
2
ffiffiffiffiffia0b
rc 1;
a0b;H
b; t
� �� �fflð33Þ
The following rule of mixture formula [14]
may be used to approximately determine the
fracture toughness for a thermally
nonhomogeneous but elastically homogeneous
ceramic-ceramic FGM with thermal parameters
graded in the x-direction:
KIcðxÞ ¼ V1ðxÞ K1Ic
ffi �2 þ V2ðxÞ K2Ic
ffi �2n o1=2
ð34Þ
in which V1(x) and V2(x) denote, respectively,
volume fractions of phases 1 and 2 and K1Ic and
K2Ic their fracture toughness.
Thermal shock damage in the FGM specimen
will be induced when the thermal shock DTexceeds DTc. The thermal shock damage may be
characterized by the arrested crack length afwhich can be determined by equating the peak
TSIF to the fracture toughness at a ¼ af with
the result
Ea0DTffiffiffiffiffiffipb
p
ð1� nÞ Max t>0f g �1
2
ffiffiffiffiffiafb
rc 1;
afb;H
b;t
� �� �¼KIcðaf Þ
ð35ÞHere the quasi-static assumption is adopted as
the inertia effect is ignored in calculating the
peak TSIF.
The thermal shock residual strength of a
ceramic-ceramic FGM is usually defined as the
fracture strength of the damaged specimen with
a crack of length af, i.e., the applied mechanical
load that causes crack initiation. For ceramic-
metal FGMs with significant rising R-curves,
the residual strength should be calculated as the
maximum applied stress during subsequent stable
crack growth. The integral equation approach can
still be used to calculate the stress intensity factor
for the damaged FGM specimen with the integral
equation having the same form as (28) and sTyyreplaced by either sa under uniform tension or by
1� 2x=bð Þsa under pure bending deformations.
T 5142 Thermal Shock Resistance of Functionally Graded Materials
The applied stress sa corresponding to the initia-
tion of periodic cracks of length af in the
FGM specimen can thus be determined by equat-
ing the SIF to the fracture toughness at a ¼ af as
follows:
KIðaf ; saÞ ¼ KIcðaf Þ ð36Þ
where KIðaf ; saÞ is the stress intensity factor for
the periodically cracked FGM plate under the
mechanical load. The stress intensity factor at
the tips of the periodic cracks in terms of the
solution of the integral equation is given by
KIðaf ; saÞ ¼ saffiffiffiffiffiffiffipaf
p � 1
2c 1;
afb;H
b
� �� �ð37Þ
The combination of (36) and (37) yields the
applied stress sa that causes crack initiation as
saðaf Þ ¼ KIcðaf Þ ffiffiffiffiffiffiffipaf
p � 1
2c 1;
afb;H
b
� � � �fflð38Þ
In general, saðaf Þ determined from (38) is
defined as the thermal shock residual strength
sR for the ceramic-ceramic FGM with periodic
edge cracks under the thermal shock DT. Forceramic-metal FGMs with significantly rising
R-curves, the residual strength is determined as
the maximum applied stress during subsequent
crack growth, i.e.,
sR ¼ Maxa>af saðaÞf g ð39Þ
Figures 5 and 6 show the critical thermal
shock and the residual tensile strength of an
alumina/silicon nitride (Al2O3/Si3N4) FGM ver-
sus thermal shock DT for various values of crack
spacing and material gradation profiles. The
reciprocal of the crack spacing can be used to
describe the crack density. The specimen thick-
ness is assumed as b¼ 5 mm, and the preexisting
surface cracks have a length a ¼ 0.05 mm
(a/b ¼ 0.01). Al2O3-coated Si3N4 cutting tools
for machining steels have been developed to
take advantage of the high-temperature defor-
mation resistance of Si3N4 and to minimize
chemical reactions of Si3N4 with steels by hav-
ing the Al2O3 coating layer. The Al2O3/Si3N4
is thus a promising candidate material for
advanced cutting tool applications. Al2O3
(95 % dense) and Si3N4 (hot pressed or sintered)
have approximately the same Young’s modulus
Thermal ShockResistance ofFunctionally GradedMaterials,Fig. 5 Thermal shock
residual bending strength of
an Al2O3/Si3N4 FGM with
an edge crack versus
thermal shock for various
values of material gradation
profile parameter p(b ¼ 5 mm, a/b ¼ 0.01)
(After Jin and Luo [10])
Thermal Shock Resistance of Functionally Graded Materials 5143 T
T
of 320 GPa [17]. Moreover, their Poisson’s
ratios are in the range of 0.2–0.28, and the dif-
ferences have insignificant effects on the frac-
ture behavior of graded materials [18]. The
material properties of the FGM are evaluated
using the three-phase micromechanics model
for conventional composites [15]. Table 1 lists
values of material parameters of Al2O3 and
Si3N4 used in the calculations. We also assume
that the volume fraction of Si3N4 follows
a simple power function
VðxÞ ¼ ðx=bÞp ð40Þ
where p is the power exponent which can be used
to describe the material gradation profile. In
numerical calculations, we only consider the
loading case of Tb ¼ T0, which means that only
the cracked surface x ¼ 0 of the FGM plate is
subjected to a temperature drop.
Figure 5 shows effects of the material grada-
tion profile (described by p in (40)) on the criticalthermal shock and the residual bending strength
of the Al2O3/Si3N4 FGMwith a single edge crack
[10]. First, the model qualitatively predicts the
residual strength behavior of quenched FGMs
observed in experiments [8], i.e., the residual
strength remains constant when the temperature
drop DT has not reached the critical thermal
shock DTc. At DT ¼ DTc, the strength suffers
a precipitous drop and then decreases gradually
with increasing severity of thermal shock.
Second, the critical shock DTc increases with
a decrease in the exponent p. For example, DTcis about 110 �C for p¼ 1.0 and increases to about
156 �C when p ¼ 0.2. Finally, the residual
strength increases with a decrease in the exponent
p. For example, the residual strength is about
54 MPa for p¼ 1.0 at DT¼ 200 �C and increases
to about 79 MPa for p ¼ 0.2. These numerical
400
350
300
250
200
150
Res
idua
l str
engt
h (M
Pa)
100
50
00 50 100 150 200 250
Thermal shock ΔT (°C)
300 350 400 450 500
Single crackH/b = 10H/b = 1H/b = 0.5
Thermal ShockResistance ofFunctionally GradedMaterials,Fig. 6 Thermal shock
residual tensile strength of
an Al2O3/Si3N4 FGM
versus thermal shock for
various values of crack
spacing H/b (p ¼ 0.2,
b ¼ 5 mm, a/b ¼ 0.01)
(After Jin and Feng [11])
Thermal ShockResistance ofFunctionally GradedMaterials,Table 1 Values of
material parameters for
Al2O3 and Si3N4
CTE
(10–6/K)
Thermal
conductivity
(W/m K)
Mass density
(kg/m3)
Specific heat
(J/kg K)
Fracture toughness
(MPa m1/2)
Al2O3 8.0 20 3,800 900 4
Si3N4 3.0 35 3,200 700 5
T 5144 Thermal Shock Resistance of Functionally Graded Materials
results indicate that the thermal shock resistance
of FGMs can be significantly enhanced with
appropriately designed material gradation pro-
files. For the Al2O3/Si3N4 FGM, the material
should transition smoothly and rapidly from
pure Al2O3 at the thermally shocked surface to
a Si3N4-rich structure to achieve optimized ther-
mal shock resistance.
Figure 6 shows effects of the crack density
(crack spacing) on the residual tensile strength
of the Al2O3/Si3N4 FGM versus thermal shock
DT. The material gradation profile parameter is
taken as p ¼ 0.2, i.e., the material is Si3N4-rich
FGM. First, the general characteristics of the
residual tensile strength behavior are similar to
those of the residual bending strength behavior
shown in Fig. 5. Second, a higher surface crack
density (smaller crack spacing H/b) enhances theresidual strength significantly. For example, at
DT ¼ 200 �C, the residual strength is about
60 MPa for a specimen with a single edge crack.
The strength is enhanced to about 135 MPa for
a periodically cracked specimen with H/b ¼ 0.5.
Finally, the residual strength gradually decreases
with increasing thermal shock severity for
a specimen having a single crack or multiple
cracks with relatively larger spacing. For
a periodically cracked specimen with H/b ¼ 0.5,
however, the strength becomes insensitive to the
severity of thermal shocks when DT is much
larger than the critical thermal shock DTc. Thisis because for a fixed ratio of crack spacing to
specimen thickness (H/b) the thermal stress
intensity factor does not change significantly
when the cracks grow longer (larger ratio of
a/H) because of more severe thermal shocks. In
fact, in the limiting case of long parallel cracks
(a/H >> 1) in a semi-infinite plate, the stress
intensity factor becomes independent of the
crack length and depends on the crack spacing
only [19]. We note that in thermal shock experi-
ments on monolithic ceramics, cracking behavior
deviates from the periodic pattern when DT is
significantly larger than DTc, which causes grad-
ual decrease in the residual strength. Finally,
it can be concluded from results presented in
Fig. 6 that for a given material gradation profile,
the crack spacing has negligible effect on
the strength for DT < DTc. This is because the
ratio of the initial crack length to the crack spac-
ing is so small (the maximum a/H is assumed as
0.02) that the stress intensity factors at the tips
of parallel cracks almost equal that for the
single crack.
References
1. Hasselman DPH (1969) Unified theory for fracture
initiation and crack propagation in brittle ceramics
subjected to thermal shock. J Am Ceram Soc
48:600–604
2. Bahr HA, Weiss HJ (1986) Heuristic approach to
thermal shock damage due to single and multiple
crack growth. Theor Appl Fract Mech 6:57–62
3. Swain MV (1990) R-curve behavior and thermal
shock resistance of ceramics. J Am Ceram Soc
73:621–628
4. Lutz EH, Swain MV (1991) Interrelation between
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strength degradation in ceramics. J Am Ceram Soc
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N (1994) Effect of microstructure scale on thermal
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materials. J Eur Ceram Soc 24:847–854
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on the residual strength behavior of thermally
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18. Delale F, Erdogan F (1983) The crack problem for
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Thermal Shock upon Thin-WalledBeams of Open Profile
Yury A. Rossikhin and Marina V. Shitikova
Research Center on Wave Dynamics in Solids
and Structures, Voronezh State University of
Architecture and Civil Engineering, Voronezh,
Russia
Synonyms
Transient dynamic response of spatially curved
thermoelastic thin-walled beam of open section
Overview
The investigation of thermally induced waves
and vibrations is known to be a very important
engineering problem [1], especially for thin-
walled beams of open section which are exten-
sively used as structural components in different
structures in civil, mechanical, and aeronautical
engineering applications, since dynamic interac-
tion between thermal fields and thin-walled solid
bodies may produce unexpected phenomena
[2–5]. The transient thermoelastic waves propa-
gating in spatially curved thin-walled beams of
generic open profile have been analyzed in [6],
wherein the dynamic theory of thin-walled beams
of open section proposed recently in [7] has
been generalized to the case of spatially curved
thermoelastic beams of open profile; the
thermoelastic features of which are described by
the Green-Naghdi [8] hyperbolic theory of
thermoelasticity without energy dissipation,
what is of great engineering importance, since
precisely curved members in modern bridges
and architectural structures continue to predomi-
nate over the straight ones because of emphasis
on aesthetics and transportation alignment
restrictions in metropolitan areas. Thus, the
increasing use of curved thin-walled beams in
highway bridges, civil engineering, and aircraft
has resulted in considerable effort that should be
directed toward developing accurate methods for
analyzing the dynamic behavior of such struc-
tures including coupling between the temperature
and strain fields.
The transient dynamic behavior of
thermoelastic spatially curved open section
beams could be analyzed using the theory of
discontinuities and the method of ray expansions
[9] (▶Ray Expansion Theory), which allow one
to find the desired fields behind the fronts of the
transient waves in terms of discontinuities in
time derivatives of the values to be found,
since these methods of solution are of frequent
use for short-time dynamic processes. Utilizing
these methods, it is possible to obtain from the
three-dimensional equations of the theory of
thermoelasticity the recurrent relationships for
determining the discontinuities in arbitrary order
time derivatives of the desired values, which, in
contrast to the Timoshenko-like theories, do not
involve shear coefficients depending on geomet-
rical parameters of thin-walled beams of
open section. This approach permits one to
solve analytically different dynamic boundary-
value problems of thermoelasticity dealing with
thermal as well as thermomechanical impact
upon a spatially curved thermoelastic thin-
walled beam of general open section, making
allowance for the beam’s translatory and rota-
tional motions, warping, rotary inertia, shear
deformation, and coupling of the temperature
and strain fields.
T 5146 Thermal Shock upon Thin-Walled Beams of Open Profile
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