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

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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

flaw resistance, R-curve behavior, and thermal shock

strength degradation in ceramics. J Am Ceram Soc

74:2859–2868

5. Cotterell B, Ong SW, Qin CD (1995) Thermal shock

and size effects in castable refractories. J Am Ceram

Soc 78:2056–2064

6. Wang H, Singh RN (1994) Thermal shock behavior of

ceramics and ceramic composites. Int Mater Rev

39:228–244

7. Schon S, Prielipp H, Janssen R, Rodel J, Claussen

N (1994) Effect of microstructure scale on thermal

shock resistance of aluminum-reinforced alumina.

J Am Ceram Soc 77:701–704

8. Zhao J, Al X, Deng HX, Wang JH (2004) Thermal

shock behaviors of functionally graded ceramic tool

materials. J Eur Ceram Soc 24:847–854

9. Jin ZH, Batra RC (1998) Thermal fracture and ther-

mal shock resistance of functionally graded materials.

In: Bahei-El-Din YA, Dvorak G (eds) Solid mechan-

ics and its applications, vol 60. Kluwer, Dordrecht,

pp 185–195

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strength of functionally graded ceramics. Mater Sci

Eng A 71–77:435–436

11. Jin ZH, Feng YZ (2008) Effects of multiple cracking

on the residual strength behavior of thermally

shocked functionally graded ceramics. Int J Solids

Struct 45:5973–5986

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the tip of an edge crack in a functionally graded

material subjected to a thermal shock. J Therm

Stresses 19:317–339

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solution of singular integral equations. In: Sih GC

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and solutions of crack problems. Noordhoff Interna-

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15. Christensen RM (1979) Mechanics of composite

materials. Wiley, New York

16. Jin ZH (2002) An asymptotic solution of temperature

field in a strip of a functionally graded material. Int

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17. Munz D, Fett T (1999) Ceramics. Springer, Berlin

18. Delale F, Erdogan F (1983) The crack problem for

a nonhomogeneous plane. J Appl Mech 50:609–614

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Publishing, Leyden, pp 131–178

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