Encyclopedia Of Thermal Stresses

Thermal Shock Resistance of Functionally Graded Materials 5137 T strength for characterizing thermal shock resistance of FGMs. The basic thermoelasticity equations 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 formulation 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 density 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.” 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 Thermoelasticity Equations of FGMs and boundary conditions. In (2)–(4), sij denote stresses, eij strains, ui displacements, dij the Kronecker delta, E(x) Young’s modulus, n(x) Poisson’s ratio, and a(x) the coefficient of thermal 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: Thermal shock behavior of FGMs is generally investigated in the standard micromechanics/ continuum framework, i.e., FGMs are treated as nonhomogeneous materials with spatially varying 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 temperature field is obtained independently of deformations. The heat conduction equation for the temperature without consideration of a heat source/sink is @ @T @T kðxÞ ¼ rðxÞcðxÞ @xi @xi @t ð1Þ sij;j ¼ 0 ð2Þ the strain–displacement relations for infinitesimal deformations eij ¼ 1ffi ui;j þ uj;i 2 ð3Þ the constitutive relation eij ¼ 1 þ nðxÞ nðxÞ sij skk dij þ aðxÞðT T 0 Þ EðxÞ EðxÞ ð4Þ sxx ¼ @2F @2F @2F ; s ¼ ; s ¼ yy xy @y2 @x2 @x@y ð5Þ 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: 1 2 @ 2 1 þn @ 2 F @ 2 1 þ n @ 2 F H F 2 E E @x2 @x2 E @y2 @y @2 1 þ n @2F ¼ H2 ½aðT T 0 Þ þ2 @x@y E @x@y where T is the temperature, t time, k(x) the spacedependent 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 2 H ð6Þ T

T 5138 Thermal Shock Resistance of Functionally Graded Materials where H2 is the Laplace operator in the xy plane. For plane strain deformations, E, n, and a are replaced by E ð1 n2 Þ, n ð1 nÞ, and ð1 þ 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. y Ta T0 x An FGM Plate with an Edge Crack Subjected to a Thermal Shock b This section describes a fracture mechanics formulation 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, m¼ m0 1 þ bðx bÞ 1 n ¼ ð 1 n0 Þ egðx bÞ 1 þ bðx bÞ a ¼ a0 eeðx bÞ ; k ¼ k0 edðx bÞ ; k ¼ k0 ð7Þ Thermal Shock Resistance of Functionally Graded Materials, Fig. 2 An FGM plate with an edge crack subjected to a thermal shock The assumed Poisson’s ratio in (7) is subjected to the constraint 0 n 0.5, and the thermal diffusivity 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 n20 2 h gðx bÞ 2 i H e H F þ H2 ½ð1 þ nÞaT ¼ 0 E0 ð10Þ ð8Þ where m is the shear modulus, k the thermal conductivity, and b, g, e, and d are material constants given by m 1 n1 b ¼ 0 1; g ¼ lnð1 þ bÞ þ ln ; m1 1 n0 a1 k1 e ¼ ln ; d ¼ ln a0 k0 T0 ð9Þ in which subscripts 0 and 1 stand for values of the parameter at x ¼ 0 and x ¼ b, respectively. H2 T þ d @T 1 @T ¼ b @x k0 @t ð11Þ 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 surface x ¼ b kept at temperature T0. The temperature distribution in the strip obtained by solving (11) is given by [12]:

Thermal Shock Resistance of Functionally Graded Materials T T 0 e d e dx ¼ DT 1 e d 1 X þ Bn e dx 2 sinðnpx Þe ðn p 2 2 þd2 4Þt 5139 T Now consider an edge crack of length a0 in the FGM strip as shown in Fig. 2. The integral equation for the cracked FGM strip is given by [12] n¼1 ð12aÞ ð1 where x* ¼ x/b, DT ¼ T0 – Ta, t ¼ tk0/b2 is the nondimensional time, and 1 1 þ kðr; sÞ ’ðsÞe ða0 bÞ½ð1þsÞ 2 g ds s r ffi 2p 1 n20 T ¼ syy ðr; tÞ; jrj E0 " # 2np 1 ð 1Þn e 3d 2 e d ð 1Þn e 3d 2 ; Bn ¼ 1 e d ð3d 2Þ2 þ n2 p2 ðd 2Þ2 þ n2 p2 where n ¼ 1;2;:::: ð12bÞ The above heat conduction problem represents an idealized thermal shock loading case, i.e., the heat transfer coefficients on the surfaces of the FGM plate are infinitely large which correspond 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 transfer coefficient. For the one-dimensional temperature field T ¼ T(x, t) given in (12), the thermal stress in the strip is given by Eayðx ; tÞ E þ 1 n ð1 n2 ÞA0 " ð1 Eayðx ; tÞ dx bðA22 x bA12 Þ 1 n sTyy ¼ Eayðx ; tÞ x dx b ðA12 x bA11 Þ 1 n # 2 0 ð13aÞ where yðx ; tÞ ¼ Tðx ; tÞ T 0 ; and constants A11, A12, A22, and A0 are defined by ðb A11 E ¼ dx; 1 n2 0 ðb A12 ¼ A21 ¼ E xdx; 1 n2 A22 ¼ E x2 dx; 1 n2 A0 ¼ A11 A22 A12 A21 0 ð13bÞ ð14Þ 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 cðrÞ ’ðrÞ ¼ eða0 bÞ½ð1þrÞ 2 g pffiffiffiffiffiffiffiffiffiffi 1 r ð16Þ where cðrÞ is continuous on [ 1, 1]. Normalizing cðrÞ by ð1 þ n0 Þa0 DT, the normalized thermal stress intensity factor (TSIF), KI , at the crack tip is obtained as rffiffi a cð1Þ b 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 K c ða0 Þ. The peak TSIF obtained from (17) is KIpeak ¼ 0 ðb @vðx; 0Þ @x ð1 n0 ÞK I 1 pffiffiffiffiffi ¼ KI ¼ 2 E0 a0 DT pb 0 ð1 ’ðxÞ ¼ 1 E0 a0 DT pffiffiffiffiffiffiffi pa0 Maxft 0g f cð1; tÞ 2g ð18Þ 1 n0 Hence, the critical thermal shock is given by DT c ¼ ð1 n0 ÞK c ða0 Þ ð19Þ pffiffiffiffiffiffiffi E0 a0 pa0 Maxft 0g f cð1; tÞ 2g T

5140 Thermal Shock Resistance of Functionally Graded Materials The corresponding critical thermal shock for the cracked ceramic specimen is DTc0 ¼ ceram ð1 n0 ÞKIc pffiffiffiffiffiffiffi E0 a0 pa0 Maxft 0g f cceram ð1; tÞ 2g ð20Þ where cceram ð1; tÞ is the solution for the 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] 1 2 DT c 1 n20 Eða0 Þ ¼ ½ 1 V m ð a0 Þ DTc0 1 n2 ða0 Þ E0 Maxft 0g f cceram ð1; tÞ 2g Maxft 0g f cð1; tÞ 2g ð21Þ where the following intrinsic fracture toughness model [14] 1 n20 EðaÞ K c ðaÞ ¼ ½1 V m ðaÞ 1 n2 ðaÞ E0 1 2 ceram KIc ð22Þ has been adopted, and Vm equals the volume fraction of the metal phase which is determined from the three-phase micromechanics model [15] and the assumed shear modulus in (7) with Vm ¼ 0 and 1 at x ¼ 0 and b, respectively. Figure 3 shows the normalized critical thermal shock DT c DTc0 versus the nondimensional initial crack length a0/b for a hypothetical FGM with (b, d, e) ¼ (1, 1, 0) [9]. It is evident that DT c 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 Cracks Subjected to a Thermal Shock This section describes a fracture mechanics formulation to calculate the critical thermal shock and the residual strength for a ceramic-ceramic FGM strip with an infinite array of periodic edge 10 Normalized critical temperature drop T 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 Normalized crack length , a0/b Thermal Shock Resistance of Functionally Graded Materials, Fig. 3 Normalized critical thermal shock versus nondimensional initial crack length (After Jin and Batra [9]) cracks subjected to quenching at the cracked surface. It also presents numerical results to illustrate 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 assumptions 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.

Thermal Shock Resistance of Functionally Graded Materials Jin [16] obtained the following closed-form, short-time asymptotic solution of the temperature 5141 field in the strip using the Laplace transform and its asymptotic properties: 1 ðx sffiffiffiffiffiffiffiffiffi kð0Þ A dx kðxÞ 0 0 1 ðb sffiffiffiffiffiffiffiffiffi T 0 T b rðbÞcðbÞkðbÞ 1 4 1 kð0Þ dxA erfc@ pffiffi kðxÞ T 0 T a rðxÞcðxÞkðxÞ 2b t Tðx; tÞ T 0 rð0Þcð0Þkð0Þ ¼ rðxÞcðxÞkðxÞ T0 Ta 1 4 T 0 1 erfc@ pffiffi 2b t ð23Þ x where kðxÞ ¼ kðxÞ rðxÞcðxÞ is the thermal diffusivity, 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 significance 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 propagate is still given by (13) with the temperature given by (23). of a point in the y-direction. Using the Fourier transform/superposition approach, the abovedescribed thermal crack problem can be reduced to finding a solution of the following singular integral equation: ð1 1 jrj 1 2pð1 n2 Þ T þ Kðr;sÞ ’ðsÞds ¼ syy ðr;tÞ; s r E 1 ð28Þ where the basic unknown is still defined in (15), nondimensional coordinates r and s are defined as 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; 1 y 1 ð24Þ sxx ¼ sxy ¼ 0; x ¼ b; 1 y 1 ð25Þ Ta T0 Tb T sxy ¼ 0; 0 x b; y ¼ nh; n ¼ 0; 1;.; 1; x v ¼ 0; 0 x b; y ¼ ð2nþ1Þh; n ¼ 0; 1;.; 1; H 2h v ¼ 0; a x b; y ¼ 2nh; n ¼ 0; 1;.; 1 ð26Þ 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 b Thermal Shock Resistance of Functionally Graded Materials, Fig. 4 An FGM plate with an array of periodic edge cracks subjected to a thermal shock

T 5142 Thermal Shock Resistance of Functionally Graded Materials 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: cðrÞ ’ðrÞ ¼ pffiffiffiffiffiffiffiffiffiffi 1 r ð30Þ where cðrÞ is a continuous and bounded function. Once (28) has been solved, the TSIF at a crack tip can be computed from ð1 nÞK I 1 pffiffiffiffiffi ¼ KI ¼ 2 Ea0 DT pb rffiffi a cð1Þ b Critical Thermal Shock and Residual Strength As stated in section “Critical Thermal Shock”, the critical thermal shock DT c 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., Maxft 0g fK I ðt; a; DT c g ¼ K Ic ða0 Þ ð31Þ where KI denotes the TSIF, KI nondimensional TSIF, DT ¼ T 0 T a , ð1 nÞK Ic ða0 Þ pffiffiffiffiffi DT c ¼ Ea0 pb 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. the and where KIc(a) is the fracture toughness of the FGM at x ¼ a. Substitution from (31) into (32) yields the critical thermal shock: ffl Maxft 0g 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: n ffi 1 2 ffi 2 2 o1 2 K Ic ðxÞ ¼ V 1 ðxÞ KIc þ V 2 ðxÞ KIc ð34Þ in which V1(x) and V2(x) denote, respectively, 1 volume fractions of phases 1 and 2 and KIc and 2 KIc their fracture toughness. Thermal shock damage in the FGM specimen will be induced when the thermal shock DT exceeds DTc. The thermal shock damage may be characterized by the arrested crack length af which can be determined by equating the peak TSIF to the fracture toughness at a ¼ af with the result ð32Þ 1 2 rffiffiffiffi a0 a0 H c 1; ; ; t b b b ð33Þ pffiffiffiffiffi rffiffiffiffi af H Ea0 DT pb 1 af Maxft 0g c 1; ; ; t ð1 nÞ 2 b b b ¼ K Ic ðaf Þ ð35Þ Here 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 ceramicmetal 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 sTyy replaced by either sa under uniform tension or by ð1 2x bÞsa 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 initiation of periodic cracks of length af in the FGM specimen can thus be determined by equating the SIF to the fracture toughness at a ¼ af as follows: K I ðaf ; sa Þ ¼ K Ic ðaf Þ ð36Þ where K I ðaf ; sa Þ 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 af H 1 pffiffiffiffiffiffiffi K I ðaf ; sa Þ ¼ sa paf c 1; ; 2 b b ð37Þ The combination of (36) and (37) yields the applied stress sa that causes crack initiation as ffl sa ðaf Þ ¼ K Ic ðaf Þ af H pffiffiffiffiffiffiffi 1 paf c 1; ; 2 b b ð38Þ In general, sa ðaf 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. For ceramic-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 fsa ðaÞg ð39Þ Figures 5 and 6 show the critical thermal shock and the residual tensile strength of an alumina/silicon nitride (Al2O3/Si3N4) FGM versus 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 thickness is assumed as b ¼ 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 deformation resistance of Si3N4 and to minimize chemical reactions of Si3N4 with steels by having 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 T

5144 Thermal Shock Resistance of Functionally Graded Materials Thermal Shock Resistance of Functionally Graded Materials, 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]) 400 Single crack H/b 10 H/b 1 H/b 0.5 350 300 Residual strength (MPa) T 250 200 150 100 50 0 0 50 100 150 200 250 300 350 400 450 500 Thermal shock ΔT ( C) Thermal Shock Resistance of Functionally Graded Materials, Table 1 Values of material parameters for Al2O3 and Si3N4 Al2O3 CTE (10–6/K) 8.0 Si3N4 3.0 Thermal conductivity (W/m K) 20 Mass density Specific heat (kg/m3) (J/kg K) 3,800 900 Fracture toughness (MPa m1/2) 4 35 3,200 5 of 320 GPa [17]. Moreover, their Poisson’s ratios are in the range of 0.2–0.28, and the differences have insignificant effects on the fracture 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. 700 Figure 5 shows effects of the material gradation profile (described by p in (40)) on the critical thermal shock and the residu

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 .