Richard B. Hetnarski Editor
Encyclopedia of Thermal Stresses
With 3310 Figures and 371 Tables Editor Professor Emeritus Richard B. Hetnarski Department of Mechanical Engineering Rochester Institute of Technology Rochester, NY, USA and Naples, FL, USA
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Wear Rate of CCBC After Thermal 5. Kaye G, Laby T (1995) Tables of physical and Treatment chemical constants, 16th edn. Longman, London 6. Santhanam AT, Tierney P, Hunt JL (1990) Cemented carbides. In: ASM handbook, vol 2 (Properties and To evaluate the effect of crack induced by selection: nonferrous alloys and special-purpose temperature changes, thermal shock experiments materials). ASM International, New Jersey were followed by erosive wear tests. Results are 7. Lanin A, Fedik I (2008) Thermal stress resistance of materials. Springer, New York 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 Further Reading cycles 1,200 C with air cooling decrease that Antonov M, Hussainova I (2006) Thermophysical negative effect of the first cycle. For instance, the properties and thermal shock resistance of chromium carbide based cermets. Proc Estonian Acad Sci Eng erosion rate after 5-th cycle 1,200 Cwithair 12(4):358–367 cooling is even decreased compared to as-received Pirso J, Valdma L, Masing J (1975) Thermal shock samples. This phenomenon may be attributed to damage resistance of cemented chromium carbide redistribution of internal stresses induced during alloys. Proc of Tallinn Tech Univ 381:39–45, (In Russian) sintering. Erosion rate of cermet with 40 wt% of Ni shows that it is less affected by temperature changes than cermet with 10 and 20 wt%.
Thermal Shock Resistance of Single Partial Immersion of CCBC Functionally Graded Materials
Partial immersion was found to be the most Zhihe Jin1 and Romesh C. Batra2 effective way of differentiation of the thermal 1Department of Mechanical Engineering, shock resistance of CCBC having various metal University of Maine, Orono, ME, USA binder content. 2Department of Engineering Science and Rectangular CCBC samples of 20 mm  Mechanics, Virginia Polytechnic Institute and 12 mm  5 mm size were polished from both State University, Blacksburg, VA, USA 20 mm  12 mm sides, heated with heating speed of 400 C minÀ1 up to 1,200 C and then immersed down to the depth of 1 mm into water Overview of room temperature. It is possible to see (Fig. 7) that the CCBC This entry introduces concepts of the critical with 40 wt% of Ni has only one thermal crack thermal shock and the thermal shock residual while cermets with low metal binder content strength for characterizing functionally graded experience multiple cracking showing their materials (FGMs). It starts with the introduction lower resistance to thermal shock. of basic heat conduction and thermoelasticity T equations for FGMs. A fracture mechanics- based formulation is then described for comput- References ing the critical thermal shock for a ceramic-metal FGM strip with an edge crack subjected to 1. Tinklepaugh JR (1960) Cermets. Reinhold, New York quenching on the cracked surface. The through- 2. Antonov M, Hussainova I (2010) Cermets surface transformation under erosive and abrasive wear. Tribol the-width variation of the shear modulus of the Int 43:1566–1575 FGM is assumed to be hyperbolic and that of 3. Thuvander M, Andren HO (2000) APFIM studies of the thermal conductivity and the coefficient grain boundaries: a review. Mater Char 44:87–100 of thermal expansion exponential. Finally a 4. Pierson H (1996) Handbook of refractory carbides and nitrides: properties, characteristics, processing, and ceramic-ceramic FGM strip with periodically applications. Noyes, New Jersey spaced surface cracks subjected to quenching is T 5136 Thermal Shock Resistance of Functionally Graded Materials considered to illustrate effects of material grada- ab s s tion and surface crack spacing on the critical ther- R R mal shock and the thermal shock residual strength.
Introduction
Functionally graded materials (FGMs) for high- temperature applications are macroscopically D D D D Tc T Tc T inhomogeneous composites usually made from ceramics and metals. The ceramic phase in an Thermal Shock Resistance of Functionally Graded FGM acts as a thermal barrier and protects the Materials, Fig. 1 Thermal shock residual strength behavior of ceramics; (a) thermal shock resistance drops metal from corrosion and oxidation, and the precipitously at DT ¼ DTc ;(b) thermal shock resistance metal phase toughens and strengthens the FGM. decreases gradually for DT > DTc High-temperature ceramic-ceramic FGMs have also been developed for cutting tools and other applications. The compositions and the volume In the first case, the strength remains unchanged fractions of constituents in an FGM are varied when the thermal shock DT is less than a critical gradually, giving a nonuniform microstructure value, DTc, called the critical thermal shock. At with continuously graded macroscopic proper- DT ¼ DTc, the strength sR suffers a precipitous ties. The knowledge of thermal shock resistance drop and then decreases gradually with an of ceramic-ceramic and ceramic-metal FGMs is increase in the severity of thermal shock. In the critical to their high-temperature applications. In second case, the strength also remains constant general, thermal shock resistance of FGMs can be for DT < DTc; however, the strength does not characterized by the critical thermal shock and drop suddenly at DT ¼ DTc but decreases gradu- thermal shock residual strength. The critical ther- ally with an increase in DT. The residual strength mal shock describes the crack initiation resis- method has been further developed to investigate tance of the material and may be determined by thermal shock behavior of monolithic ceramics in equating the fracture toughness to the peak the context of thermo-fracture mechanics (see, thermal stress intensity factor at the tip of e.g., [2–5]). a preexisting crack emanating from the surface The critical thermal shock and residual and going into the FGM body. The thermal shock strength methods have been employed to residual strength is a damage tolerance property evaluate thermal shock resistance of ceramic describing the load carrying capacity of a struc- composites in recent years. Examples include ture damaged with a thermal shock. experimental investigations on fiber-reinforced The residual strength method to find the ther- ceramic matrix composites [6], metal mal fracture resistance of monolithic ceramics particulate-reinforced ceramic matrix composites was developed by Hasselman [1]. Micro-cracks [7], and ceramic-ceramic FGMs [8]. These exper- inherently exist in ceramics. When a ceramic imental studies showed that the residual strength specimen is subjected to sufficiently severe ther- method is an effective and convenient approach mal shocks, some of the preexisting micro-cracks for evaluating thermal shock resistance of will grow to form macro-cracks. Crack propaga- ceramic composites. Jin and Batra [9], Jin and tion in thermally shocked ceramics may be Luo [10], and Jin and Feng [11] developed theo- arrested depending on the severity of the thermal retical thermo-fracture mechanics models to shock, thermal stress field characteristics, and evaluate the critical thermal shock and thermal material properties. The measured strength of shock residual strength of FGMs. a thermally shocked ceramic specimen generally This entry introduces concepts of the critical exhibits two kinds of behavior as shown in Fig. 1. thermal shock and the thermal shock residual Thermal Shock Resistance of Functionally Graded Materials 5137 T strength for characterizing thermal shock resis- index. Equation 1 has been written in rectangular tance of FGMs. The basic thermoelasticity equa- Cartesian coordinates (x1, x2, x3) which we will tions of FGMs are described in section sometimes also denote by (x, y, z). “Thermoelasticity Equations of FGMs.” The basic equations of thermoelasticity Section “An FGM Plate with an Edge Crack include the equations of equilibrium in the Subjected to a Thermal Shock” considers absence of body forces a ceramic-metal FGM strip with an edge crack subjected to quenching on the cracked surface sij;j ¼ 0 ð2Þ and derives a fracture mechanics-based formula- tion to determine the critical thermal shock. the strain–displacement relations for infinitesi- Section “An FGM Plate with Parallel Edge mal deformations Cracks Subjected to a Thermal Shock” considers a ceramic-ceramic FGM strip with periodically 1 ffiÁ e ¼ u ; þ u ; ð3Þ spaced surface cracks subjected to quenching. ij 2 i j j i The effects of material gradation and crack den- sity on the critical thermal shock and the thermal the constitutive relation shock residual strength are also examined for an Al O /Si N FGM plate in section “An FGM 2 3 3 4 1 þ nðxÞ nðxÞ Plate with Parallel Edge Cracks Subjected to e ¼ s À s d þ aðxÞðÞT À T0 ij EðxÞ ij EðxÞ kk ij a Thermal Shock.” ð4Þ
Thermoelasticity Equations of FGMs and boundary conditions. In (2)–(4), sij denote stresses, eij strains, ui displacements, dij the Thermal shock behavior of FGMs is generally Kronecker delta, E(x) Young’s modulus, n(x) investigated in the standard micromechanics/ Poisson’s ratio, and a(x) the coefficient of ther- continuum framework, i.e., FGMs are treated as mal expansion, and the FGM has been assumed to nonhomogeneous materials with spatially vary- be isotropic. A comma followed by index j ing thermomechanical properties that are found implies partial derivative with respect to xj. by using the conventional micromechanics Under plane stress conditions, the equilibrium models for homogenizing material properties of equations can be satisfied by expressing stresses composites. Moreover, an uncoupled approach is in terms of the Airy stress function F as follows: adopted in which the influence of deformation on temperature is ignored, and hence the tempera- @2F @2F @2F s ¼ ; s ¼ ; s ¼À ð5Þ ture field is obtained independently of deforma- xx @y2 yy @x2 xy @x@y tions. The heat conduction equation for the temperature without consideration of a heat Use of the constitutive relation (4) and the T source/sink is strain compatibility conditions derived from (3) yields the following governing equation for the ! Airy stress function for general nonhomogeneous @ @T @T kðxÞ ¼ rðxÞcðxÞ ð1Þ materials: @xi @xi @t 1 @2 1þn @2F @2 1þn @2F where T is the temperature, t time, k(x) the space- H2 H2F À À E @y2 E @x2 @x2 E @y2 dependent thermal conductivity, r(x) the mass @2 n @2 density, and c(x) the specific heat. The Latin 1þ F 2 þ2@ @ @ @ ¼ÀH ½aðÞT ÀT0 indices have the range 1, 2, and 3, and repeated x y E x y indices imply summation over the range of the ð6Þ T 5138 Thermal Shock Resistance of Functionally Graded Materials where H2 is the Laplace operator in the xy y plane. For plane strain deformations, E, n, and a are replaced by E=ð1 À n2Þ, n=ð1 À nÞ, and ð1 þ nÞa, respectively. In analysis of deformations of FGMs, E and n and other material parameters are assumed to be Ta T0 T0 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 x with the micromechanics analyses.
An FGM Plate with an Edge Crack Subjected to a Thermal Shock b
This section describes a fracture mechanics for- Thermal Shock Resistance of Functionally Graded mulation to calculate the critical thermal shock Materials, Fig. 2 An FGM plate with an edge crack for a ceramic-metal FGM strip of width b with an subjected to a thermal shock edge crack subjected to quenching on the cracked surface [9]; e.g., see Fig. 2. The assumed Poisson’s ratio in (7) is subjected to Basic Equations the constraint 0 n < 0.5, and the thermal diffu- The through-the-width variation of the shear sivity is assumed to be constant for mathematical modulus is assumed to be hyperbolic and that of convenience. This can be achieved by suitably the thermal conductivity and the coefficient of varying the specific heat. thermal expansion exponential. That is, With (7)–(9) and the assumption that a plane strain state of deformation prevails in the body, m (6) and (1) can be written as m ¼ 0 1 þ bðx=bÞ gðx=bÞ ð7Þ 2 hi e 1 À n0 2 gðx=bÞ 2 2 1 À n ¼ ðÞ1 À n0 H e H F þ H ½¼ð1 þ nÞaT 0 1 þ bðx=bÞ E0 ð10Þ eðx=bÞ dðx=bÞ a ¼ a0e ; k ¼ k0e ; k ¼ k0 ð8Þ d @T 1 @T H2T þ ¼ ð11Þ where m is the shear modulus, k the thermal b @x k0 @t conductivity, and b, g, e, and d are material con- stants given by Temperature, Thermal Stress, and Thermal m 1 À n1 Stress Intensity Factor b ¼ 0 À 1; g ¼ lnð1 þ bÞþln ; m 1 À n Assume that the FGM strip is initially at 1 0 ð9Þ a k a uniform temperature T , the surface x ¼ 0is e ¼ ln 1 ; d ¼ ln 1 0 a0 k0 suddenly cooled to temperature Ta with the sur- face x ¼ b kept at temperature T0. The tempera- in which subscripts 0 and 1 stand for values of ture distribution in the strip obtained by solving the parameter at x ¼ 0 and x ¼ b, respectively. (11) is given by [12]: Thermal Shock Resistance of Functionally Graded Materials 5139 T
d dxà T À T eÀ À eÀ Now consider an edge crack of length a0 0 ¼ DT 1 À eÀd in the FGM strip as shown in Fig. 2. The X1 integral equation for the cracked FGM strip is ÀdxÃ=2 à Àðn2p2þd2=4Þt þ Bne sinðnpx Þe given by [12] n¼1 ð12aÞ ð1 ! 1 = = þ kðr; sÞ ðsÞeÀða0 bÞ½ð1þsÞ 2gds s À r where x* ¼ x/b, DT ¼ T0 – Ta, t ¼ tk0/b2 is the À1 ffiÁ nondimensional time, and 2p 1 À n2 0 T ; ; "#¼À syyðr tÞ jrj1 ð14Þ n = n = E 2np 1ÀðÀ1Þ eÀ3d 2 eÀd ÀðÀ1Þ eÀ3d 2 0 Bn ¼ À ; 1ÀeÀd ð3d=2Þ2 þn2p2 ðd=2Þ2 þn2p2 where n ¼ 1;2;:::: @vðx; 0Þ ð12bÞ ðxÞ¼ ð15Þ @x The above heat conduction problem repre- sents an idealized thermal shock loading case, with v(x, 0) being the displacement in the i.e., the heat transfer coefficients on the surfaces y-direction at the crack surface, and k(r, s)isa of the FGM plate are infinitely large which cor- known kernel. According to the singular equation respond to the severest thermal stress induced in theory [13], (14) has a solution of the form the plate. In other words, the critical thermal shock predicted by the current model would be = = cðrÞ ðrÞ¼eða0 bÞ½ð1þrÞ 2g pffiffiffiffiffiffiffiffiffiffiffi ð16Þ lower than that obtained using a finite heat trans- 1 À r fer coefficient. For the one-dimensional temper- ature field T ¼ T(x, t) given in (12), the thermal where cðrÞ is continuous on [À1, 1]. Normaliz- stress in the strip is given by ing cðrÞ by ð1 þ n0Þa0DT, the normalized ther- mal stress intensity factor (TSIF), KÃ, at the crack EayðxÃ; tÞ E I sT ¼À þ tip is obtained as yy 1 À n ðÞ1 À n2 A 0 rffiffiffi " ð1 Ã; à ð1 À n0ÞKI 1 a à Eayðx tÞ Ã K ¼ pffiffiffiffiffiffi ¼À cð1Þð17Þ Â bAðÞ22 À x bA12 dx I 1 À n E0a0DT pb 2 b 0 # ð1 The value of the TSIF can be computed once EayðxÃ; tÞ À b2ðÞA À xÃbA xÃdxà (14) has been solved. 12 11 1 À n 0 Critical Thermal Shock ð13aÞ The TSIF in (17) is a function of time. The critical T à à where yðx ; tÞ¼Tðx ; tÞÀT0; and constants thermal shock may be obtained by equating the A11, A12, A22, and A0 are defined by peak TSIF to the intrinsic fracture toughness Kcða0Þ. The peak TSIF obtained from (17)is ðb ðb E E E a DT pffiffiffiffiffiffiffi A11 ¼ dx; A12 ¼ A21 ¼ xdx; peak 0 0 ; = n2 n2 KI ¼ pa0Maxfgt>0 fgÀcð1 tÞ 2 ð18Þ 1 À 1 À 1 À n0 0 0 ðb E Hence, the critical thermal shock is given by A ¼ x2dx; A ¼ A A À A A 22 1 À n2 0 11 22 12 21 1 n K a 0 ffiffiffiffiffiffiffiðÞÀ 0 cð 0Þ DTc ¼ p ; = ð19Þ ð13bÞ E0a0 pa0Maxfgt>0 fgÀcð1 tÞ 2 T 5140 Thermal Shock Resistance of Functionally Graded Materials
The corresponding critical thermal shock for 10 the cracked ceramic specimen is
ðÞ1 À n Kceram 8 D 0 ffiffiffiffiffiffiffi 0 Ic Tc ¼ p ; = E0a0 pa0Maxfgt>0 fgÀcceramð1 tÞ 2 ð20Þ 6 where c ð1; tÞ is the solution for the ceram 4 corresponding crack problem of a ceramic strip ceram and KIc is the fracture toughness of the ceramic. It follows from (19) and (20) that [9] 2 Normalized critical temperature drop &' 2 1=2 DTc 1 À n Eða0Þ 0 0 0 0.1 0.2 0.3 0.4 0.5 0 ¼ ½1 À VmðÞa0 2 DT 1 À n ða0Þ E0 c Normalized crack length , a0/b ; = Maxfgt>0 fgÀcceramð1 tÞ 2 Thermal Shock Resistance of Functionally Graded Max > fgÀcð1; tÞ=2 fgt 0 Materials, Fig. 3 Normalized critical thermal shock ver- ð21Þ sus nondimensional initial crack length (After Jin and Batra [9]) where the following intrinsic fracture toughness model [14] cracks subjected to quenching at the cracked sur- &'face. It also presents numerical results to illus- 2 1=2 1 À n0 EðaÞ ceram trate effects of the material gradation profile and KcðaÞ¼ ½1 À VmðaÞ 2 KIc 1 À n ðaÞ E0 the surface crack density on the thermal shock ð22Þ resistance of the FGM strip [11]. The FGM is assumed to have constant Young’s modulus and has been adopted, and Vm equals the volume frac- Poisson’s ratio but arbitrarily graded thermal tion of the metal phase which is determined from properties along the width. While these assump- the three-phase micromechanics model [15]and tions limit the application of the model, there the assumed shear modulus in (7) with Vm ¼ 0and exist FGM systems, e.g., TiC/SiC, MoSi2/ 3 1atx ¼ 0andb,respectively.Figure shows the Al2O3, and Al2O3/Si3N4, for which Young’s = 0 normalized critical thermal shock DTc DTc ver- modulus is nearly a constant. sus the nondimensional initial crack length a0/b for a hypothetical FGM with (b, d, e) ¼ (1, 1, 0) [9]. It Temperature and Thermal Stress Fields is evident that DTc for the FGM is significantly We consider an infinitely long ceramic-ceramic higher than that for the ceramic. Hence, the FGM strip of width b with an infinite array of cracked FGM strip can withstand a more severe periodic edge cracks of length a and spacing thermal shock than the corresponding ceramic between cracks H ¼ 2 h as shown in Fig. 4. The strip without the crack propagating into the strip. thermal parameters of the FGM are arbitrarily graded in the width (x-) direction. The strip is
initially at a constant temperature T0, and its An FGM Plate with Parallel Edge Cracks surfaces x ¼ 0 and x ¼ b are suddenly cooled to Subjected to a Thermal Shock temperatures Ta and Tb, respectively. Since the bounding surfaces x ¼ 0 and x ¼ b are kept at This section describes a fracture mechanics for- uniform temperatures, and material gradation and mulation to calculate the critical thermal shock cracking are in the x-direction, it is reasonable and the residual strength for a ceramic-ceramic to assume that the heat for short times flows in the FGM strip with an infinite array of periodic edge x-direction. Thermal Shock Resistance of Functionally Graded Materials 5141 T
Jin [16] obtained the following closed-form, field in the strip using the Laplace transform and short-time asymptotic solution of the temperature its asymptotic properties:
0 sffiffiffiffiffiffiffiffiffi 1 != ðx Tðx; tÞÀT rð0Þcð0Þkð0Þ 1 4 1 kð0Þ 0 ¼À erfc@ pffiffiffi dxA T0 À Ta rðxÞcðxÞkðxÞ 2b t kðxÞ 0 0 sffiffiffiffiffiffiffiffiffi 1 ð23Þ != ðb T À T rðbÞcðbÞkðbÞ 1 4 1 kð0Þ À 0 b erfc@ pffiffiffi dxA T0 À Ta rðxÞcðxÞkðxÞ 2b t kðxÞ x where kðxÞ¼kðxÞ=rðxÞcðxÞ is the thermal diffu- of a point in the y-direction. Using the Fourier sivity, t ¼ kð0Þt=b2 is the nondimensional time, transform/superposition approach, the above- and erfc( ) is the complementary error function. described thermal crack problem can be reduced The asymptotic solution given by (23) holds for to finding a solution of the following singular an FGM plate with continuous and piecewise integral equation: differentiable thermal parameters. The signifi- cance of the solution lies in the fact that the ð1 ! 1 2pð1Àn2Þ thermal stress and the thermal stress intensity þKðr;sÞ ðsÞds¼À sT ðr;tÞ; sÀr E yy factor (TSIF) in the FGM plate induced by the À1 thermal shock reach their peak values in a very jrj1 short time. Thus, (23) may be used to evaluate ð28Þ peak values of the thermal stress and the TSIF which govern the failure of the material. The where the basic unknown is still defined in (15), thermal stress that causes edge cracks to propa- nondimensional coordinates r and s are defined as gate is still given by (13) with the temperature given by (23). y 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; À1 < y < 1ð24Þ Ta T0 Tb
sxx ¼ sxy ¼ 0; x ¼ b; À1 < y < 1ð25Þ T sxy ¼0; 0 T syy ¼Às ; 0 < x < a; y ¼ 2nh; b yy ð27Þ n ¼ 0; Æ1; ...; Æ1 Thermal Shock Resistance of Functionally Graded T where syy is given by (13) with the temperature Materials, Fig. 4 An FGM plate with an array of peri- given by (23), and v equals the displacement odic edge cracks subjected to a thermal shock T 5142 Thermal Shock Resistance of Functionally Graded Materials 0 r ¼ 2x=a À 1; s ¼ 2x =a À 1 ð29Þ a0 ¼ að0Þ.In(31), cð1Þ is a function of nondimensional time t, the nondimensional and K(r, s) is a known kernel [11]. crack length a/b, the crack spacing According to the singular integral equation parameter H/b, and the material gradation theory [13], the solution of (28) has the parameter. following form: Critical Thermal Shock and Residual Strength cðrÞ As stated in section “Critical Thermal Shock”, ðrÞ¼pffiffiffiffiffiffiffiffiffiffiffi ð30Þ 1 À r the critical thermal shock DTc that causes the initiation of the parallel cracks of length a may where cðrÞ is a continuous and bounded function. be obtained by equating the peak TSIF to the Once (28) has been solved, the TSIF at a crack tip fracture toughness of the FGM, i.e., can be computed from rffiffiffi ; ; Maxfgt>0 fg¼KIðt a DTc KIcða0Þð32Þ Ã ð1 À nÞKffiffiffiffiffiffiI 1 a KI ¼ p ¼À cð1Þð31Þ Ea0DT pb 2 b where KIc(a) is the fracture toughness of the FGM à where KI denotes the TSIF, KI the at x ¼ a. Substitution from (31) into (32) yields nondimensional TSIF, DT ¼ T0 À Ta, and the critical thermal shock: ffl &'rffiffiffiffiffi ð1 À nÞKffiffiffiffiffiffiIcða0Þ 1 a0 ; a0 ; H ; DTc¼ p Maxfgt>0 À c 1 t ð33Þ Ea0 pb 2 b b b pffiffiffiffiffiffi &'rffiffiffiffiffi The following rule of mixture formula [14] Ea0DT pb 1 af ;af ;H ; Maxfgt>0 À c 1 t may be used to approximately determine the ð1 À nÞ 2 b b b fracture toughness for a thermally ¼ KIcðaf Þ nonhomogeneous but elastically homogeneous ð35Þ ceramic-ceramic FGM with thermal parameters graded in the x-direction: Here the quasi-static assumption is adopted as the inertia effect is ignored in calculating the no ffiÁ ffiÁ1=2 peak TSIF. 1 2 2 2 KIcðxÞ¼ V1ðxÞ KIc þ V2ðxÞ KIc The thermal shock residual strength of a ceramic-ceramic FGM is usually defined as the ð34Þ fracture strength of the damaged specimen with a crack of length af, i.e., the applied mechanical in which V1(x) and V2(x) denote, respectively, load that causes crack initiation. For ceramic- 1 volume fractions of phases 1 and 2 and KIc and metal FGMs with significant rising R-curves, 2 KIc their fracture toughness. the residual strength should be calculated as the Thermal shock damage in the FGM specimen maximum applied stress during subsequent stable will be induced when the thermal shock DT crack growth. The integral equation approach can exceeds DTc. The thermal shock damage may be still be used to calculate the stress intensity factor characterized by the arrested crack length af for the damaged FGM specimen with the integral T which can be determined by equating the peak equation having the same form as (28) and syy TSIF to the fracture toughness at a ¼ af with replaced by either sa under uniform tension or by the result ðÞ1 À 2x=b sa under pure bending deformations. Thermal Shock Resistance of Functionally Graded Materials 5143 T Thermal Shock Resistance of Functionally Graded Materials, 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]) The applied stress sa corresponding to the initia- sR for the ceramic-ceramic FGM with periodic tion of periodic cracks of length af in the edge cracks under the thermal shock DT. For FGM specimen can thus be determined by equat- ceramic-metal FGMs with significantly rising ing the SIF to the fracture toughness at a ¼ af as R-curves, the residual strength is determined as follows: the maximum applied stress during subsequent crack growth, i.e., KIðaf ; saÞ¼KIcðaf Þð36Þ sR ¼ Maxa>af fgsaðaÞ ð39Þ where KIðaf ; saÞ is the stress intensity factor for the periodically cracked FGM plate under the Figures 5 and 6 show the critical thermal mechanical load. The stress intensity factor at shock and the residual tensile strength of an the tips of the periodic cracks in terms of the alumina/silicon nitride (Al2O3/Si3N4)FGMver- solution of the integral equation is given by sus thermal shock DT for various values of crack spacing and material gradation profiles. The &'reciprocal of the crack spacing can be used to pffiffiffiffiffiffiffi 1 af H KIðaf ; saÞ¼sa paf À c 1; ; describe the crack density. The specimen thick- 2 b b ness is assumed as b ¼ 5 mm, and the preexisting T ð37Þ surface cracks have a length a ¼ 0.05 mm (a/b ¼ 0.01). Al2O3-coated Si3N4 cutting tools The combination of (36) and (37) yields the for machining steels have been developed to applied stress sa that causes crack initiation as take advantage of the high-temperature defor- ffl&' !mation resistance of Si3N4 and to minimize pffiffiffiffiffiffiffi 1 af H chemical reactions of Si3N4 with steels by hav- saðaf Þ¼KIcðaf Þ paf À c 1; ; 2 b b ing the Al2O3 coating layer. The Al2O3/Si3N4 ð38Þ is thus a promising candidate material for advanced cutting tool applications. Al2O3 In general, saðaf Þ determined from (38)is (95 % dense) and Si3N4 (hot pressed or sintered) defined as the thermal shock residual strength have approximately the same Young’s modulus T 5144 Thermal Shock Resistance of Functionally Graded Materials Thermal Shock 400 Resistance of Single crack Functionally Graded 350 H/b = 10 Materials, H/b = 1 Fig. 6 Thermal shock H/b = 0.5 residual tensile strength of 300 an Al2O3/Si3N4 FGM versus thermal shock for 250 various values of crack spacing H/b (p ¼ 0.2, b ¼ 5 mm, a/b ¼ 0.01) 200 (After Jin and Feng [11]) 150 Residual strength (MPa) 100 50 0 0 50 100 150 200 250 300 350 400 450 500 Thermal shock ΔT (°C) Thermal Shock Thermal Resistance of CTE conductivity Mass density Specific heat Fracture toughness Functionally Graded (10–6/K) (W/m K) (kg/m3) (J/kg K) (MPa m1/2) Materials, Table 1 Values of Al2O3 8.0 20 3,800 900 4 material parameters for Si3N4 3.0 35 3,200 700 5 Al2O3 and Si3N4 of 320 GPa [17]. Moreover, their Poisson’s Figure 5 shows effects of the material grada- ratios are in the range of 0.2–0.28, and the dif- tion profile (described by p in (40)) on the critical ferences have insignificant effects on the frac- thermal shock and the residual bending strength ture behavior of graded materials [18]. The of the Al2O3/Si3N4 FGM with a single edge crack material properties of the FGM are evaluated [10]. First, the model qualitatively predicts the using the three-phase micromechanics model residual strength behavior of quenched FGMs for conventional composites [15]. Table 1 lists observed in experiments [8], i.e., the residual values of material parameters of Al2O3 and strength remains constant when the temperature Si3N4 used in the calculations. We also assume drop DT has not reached the critical thermal that the volume fraction of Si3N4 follows shock DTc.AtDT ¼ DTc, the strength suffers asimplepowerfunction a precipitous drop and then decreases gradually with increasing severity of thermal shock. p VðxÞ¼ðx=bÞ ð40Þ Second, the critical shock DTc increases with a decrease in the exponent p. For example, DTc where p is the power exponent which can be used is about 110 C for p ¼ 1.0 and increases to about to describe the material gradation profile. In 156 C when p ¼ 0.2. Finally, the residual numerical calculations, we only consider the strength increases with a decrease in the exponent loading case of Tb ¼ T0, which means that only p. For example, the residual strength is about the cracked surface x ¼ 0 of the FGM plate is 54 MPa for p ¼ 1.0 at DT ¼ 200 C and increases subjected to a temperature drop. to about 79 MPa for p ¼ 0.2. These numerical Thermal Shock Resistance of Functionally Graded Materials 5145 T results indicate that the thermal shock resistance the strength for DT < DTc. This is because the of FGMs can be significantly enhanced with ratio of the initial crack length to the crack spac- appropriately designed material gradation pro- ing is so small (the maximum a/H is assumed as files. For the Al2O3/Si3N4 FGM, the material 0.02) that the stress intensity factors at the tips should transition smoothly and rapidly from of parallel cracks almost equal that for the pure Al2O3 at the thermally shocked surface to single crack. aSi3N4-rich structure to achieve optimized ther- mal shock resistance. Figure 6 shows effects of the crack density (crack spacing) on the residual tensile strength References of the Al2O3/Si3N4 FGM versus thermal shock DT. The material gradation profile parameter is 1. Hasselman DPH (1969) Unified theory for fracture initiation and crack propagation in brittle ceramics taken as p ¼ 0.2, i.e., the material is Si3N4-rich subjected to thermal shock. J Am Ceram Soc FGM. First, the general characteristics of the 48:600–604 residual tensile strength behavior are similar to 2. Bahr HA, Weiss HJ (1986) Heuristic approach to those of the residual bending strength behavior thermal shock damage due to single and multiple shown in Fig. 5. Second, a higher surface crack crack growth. Theor Appl Fract Mech 6:57–62 3. Swain MV (1990) R-curve behavior and thermal density (smaller crack spacing H/b) enhances the shock resistance of ceramics. J Am Ceram Soc residual strength significantly. For example, at 73:621–628 DT ¼ 200 C, the residual strength is about 4. Lutz EH, Swain MV (1991) Interrelation between 60 MPa for a specimen with a single edge crack. flaw resistance, R-curve behavior, and thermal shock strength degradation in ceramics. J Am Ceram Soc The strength is enhanced to about 135 MPa for 74:2859–2868 a periodically cracked specimen with H/b ¼ 0.5. 5. Cotterell B, Ong SW, Qin CD (1995) Thermal shock Finally, the residual strength gradually decreases and size effects in castable refractories. J Am Ceram with increasing thermal shock severity for Soc 78:2056–2064 6. Wang H, Singh RN (1994) Thermal shock behavior of a specimen having a single crack or multiple ceramics and ceramic composites. Int Mater Rev cracks with relatively larger spacing. For 39:228–244 a periodically cracked specimen with H/b ¼ 0.5, 7. Schon S, Prielipp H, Janssen R, Rodel J, Claussen however, the strength becomes insensitive to the N (1994) Effect of microstructure scale on thermal D shock resistance of aluminum-reinforced alumina. severity of thermal shocks when T is much J Am Ceram Soc 77:701–704 larger than the critical thermal shock DTc.This 8. Zhao J, Al X, Deng HX, Wang JH (2004) Thermal is because for a fixed ratio of crack spacing to shock behaviors of functionally graded ceramic tool specimen thickness (H/b) the thermal stress materials. J Eur Ceram Soc 24:847–854 9. Jin ZH, Batra RC (1998) Thermal fracture and ther- intensity factor does not change significantly mal shock resistance of functionally graded materials. when the cracks grow longer (larger ratio of In: Bahei-El-Din YA, Dvorak G (eds) Solid mechan- a/H) because of more severe thermal shocks. In ics and its applications, vol 60. Kluwer, Dordrecht, fact, in the limiting case of long parallel cracks pp 185–195 T >> 10. Jin ZH, Luo WJ (2006) Thermal shock residual (a/H 1) in a semi-infinite plate, the stress strength of functionally graded ceramics. Mater Sci intensity factor becomes independent of the Eng A 71–77:435–436 crack length and depends on the crack spacing 11. Jin ZH, Feng YZ (2008) Effects of multiple cracking only [19]. We note that in thermal shock experi- on the residual strength behavior of thermally shocked functionally graded ceramics. Int J Solids ments on monolithic ceramics, cracking behavior Struct 45:5973–5986 deviates from the periodic pattern when DT is 12. Jin ZH, Batra RC (1996) Stress intensity relaxation at significantly larger than DTc, which causes grad- the tip of an edge crack in a functionally graded ual decrease in the residual strength. Finally, material subjected to a thermal shock. J Therm Stresses 19:317–339 it can be concluded from results presented in 13. Erdogan F, Gupta GD, Cook TS (1973) Numerical Fig. 6 that for a given material gradation profile, solution of singular integral equations. In: Sih GC the crack spacing has negligible effect on (ed) Mechanics of fracture, vol 1, Methods of analysis T 5146 Thermal Shock upon Thin-Walled Beams of Open Profile and solutions of crack problems. Noordhoff Interna- wherein the dynamic theory of thin-walled beams tional Publishing, Leyden, pp 368–425 of open section proposed recently in [7] has 14. Jin ZH, Batra RC (1996) Some basic fracture mechan- ics concepts in functionally graded materials. J Mech been generalized to the case of spatially curved Phys Solids 44:1221–1235 thermoelastic beams of open profile; the 15. Christensen RM (1979) Mechanics of composite thermoelastic features of which are described by materials. Wiley, New York the Green-Naghdi [8] hyperbolic theory of 16. Jin ZH (2002) An asymptotic solution of temperature field in a strip of a functionally graded material. Int thermoelasticity without energy dissipation, Commun Heat Mass Transfer 29:887–895 what is of great engineering importance, since 17. Munz D, Fett T (1999) Ceramics. Springer, Berlin precisely curved members in modern bridges 18. Delale F, Erdogan F (1983) The crack problem for and architectural structures continue to predomi- a nonhomogeneous plane. J Appl Mech 50:609–614 19. Benthem JP, Koiter WT (1973) Asymptotic approxi- nate over the straight ones because of emphasis mations to crack problems. In: Sih GC (ed) Mechan- on aesthetics and transportation alignment ics of fracture, vol 1, Methods of analysis and restrictions in metropolitan areas. Thus, the solutions of crack problems. Noordhoff International increasing use of curved thin-walled beams in Publishing, Leyden, pp 131–178 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- Thermal Shock upon Thin-Walled tures including coupling between the temperature Beams of Open Profile and strain fields. The transient dynamic behavior of Yury A. Rossikhin and Marina V. Shitikova thermoelastic spatially curved open section Research Center on Wave Dynamics in Solids beams could be analyzed using the theory of and Structures, Voronezh State University of discontinuities and the method of ray expansions Architecture and Civil Engineering, Voronezh, [9](▶ Ray Expansion Theory), which allow one Russia 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, Synonyms since these methods of solution are of frequent use for short-time dynamic processes. Utilizing Transient dynamic response of spatially curved these methods, it is possible to obtain from the thermoelastic thin-walled beam of open section three-dimensional equations of the theory of thermoelasticity the recurrent relationships for determining the discontinuities in arbitrary order Overview time derivatives of the desired values, which, in contrast to the Timoshenko-like theories, do not The investigation of thermally induced waves involve shear coefficients depending on geomet- and vibrations is known to be a very important rical parameters of thin-walled beams of engineering problem [1], especially for thin- open section. This approach permits one to walled beams of open section which are exten- solve analytically different dynamic boundary- sively used as structural components in different value problems of thermoelasticity dealing with structures in civil, mechanical, and aeronautical thermal as well as thermomechanical impact engineering applications, since dynamic interac- upon a spatially curved thermoelastic thin- tion between thermal fields and thin-walled solid walled beam of general open section, making bodies may produce unexpected phenomena allowance for the beam’s translatory and rota- [2–5]. The transient thermoelastic waves propa- tional motions, warping, rotary inertia, shear gating in spatially curved thin-walled beams of deformation, and coupling of the temperature generic open profile have been analyzed in [6], andstrainfields.