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Effects of substrate thickness and heat transfer scheme on edge cracking of a brittle coating due to thermal transients
Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
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Effects of substrate thickness and heat transfer scheme on edge cracking of a brittle coating due to thermal transients Xuejun Chen a,b,⇑, Xiuping Wang a a b
Department of Applied Mechanics, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, China Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 25 October 2016 Revised 2 March 2017 Accepted 8 March 2017 Available online 9 March 2017 Keywords: Coating Edge cracking Weight function Thermal stress intensity factor (TSIF)
a b s t r a c t Brittle coatings may suffer edge cracking due to thermal transients. The thermal stress intensity factor (TSIF) is used here to characterize the driving force for such damage modes. The purpose of this paper is to investigate the effect of substrate thickness and heat transfer scheme on the edge cracking behavior. The transient temperature and thermal stress field of the un-cracked system are first obtained in closed forms. The weight functions suitable for thermal loading are then developed for the edge-cracked coating. The TSIF at the crack tip is finally obtained based on the principle of superposition, with the equal thermal stresses utilized as the crack surface tractions. The dependence of the normalized TSIF is examined on different thermal boundary condition, normalized time, and relative crack depth as well as substrate/coating thickness ratio. It is found the thicker substrate leads to much higher driving force for the edge cracking. The heat transfer schemes on the lower surface of the substrate, however, have insignificant effect on it. The findings of this study may assist in the integrity analysis of coatings due to rapidly changing thermal environments. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Protective coatings are usually deposited onto underlying substrates in a broad range of technological applications, either to enhance certain mechanical properties of substrate materials or to endow the surface with new properties (e.g. hardness, corrosion and thermal resistance) [1–3]. In most cases, such thin coatings need to bear very high stresses caused, for example, by mechanical loads, temperature gradients or other sources. Consequently, once the in-plane stress reaches a critical condition, edge cracks may be produced near the surface [1,2]. Under repeated loading, these cracks would then result in more serious interfacial delamination and final removal of coating upon further extending toward and near the interface (see Fig. 1). Even if the coating spallation were not induced, edge cracks touching on the interface may still act as convenient tunnels for corrosive species to flow in and damage the substrate and interface. This scenario regularly occurs in corrosive environments, a typical example of which can be found in a Cr coated gun barrel during the firing process [3,4]. Therefore, a major challenge facing the coating-substrate system lies with that the ⇑ Corresponding author at Department of Applied Mechanics, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, China. E-mail addresses: [email protected] (X. Chen), [email protected] (X. Wang). http://dx.doi.org/10.1016/j.tafmec.2017.03.006 0167-8442/Ó 2017 Elsevier Ltd. All rights reserved.
coating shall be able to remain intact and adhered to substrate throughout the service period. One way to retain the integrity of coating is to delay or avoid edge cracking. Hence various concepts of microstructure design were proposed in materials community such as duplex coating, multilayer coating as well as functionally gradient coating [5–7]. The goal for such designs is to alleviate the misfit strain between the coating and substrate and ultimately to reduce the resulting stress level. In parallel efforts, a lot of processing techniques have been recently developed, the intention of which is to toughen coatings or the interface, and thereby to enhance their crack growth resistance [8,9]. From the point view of fracture mechanics, it can also be achieved by reducing the driving force for edge crack initiation and propagation. This requires the stress intensity factor (SIF) of edge cracks be quantified as a function of crack depth, externally applied loads and material constants as well. By knowing what and how main parameters affect the SIFs, one can make a reliable prediction of crack growth rates and also an optimal selection of associated quantities for cracking suppression. In this paper, we focus on an edge cracking of a brittle coating due to thermal transients. A lot of techniques (e.g. method of singular integral equations, finite element method) have been developed to estimate the SIFs for an edge crack in coatings [10–13]. Most of them, however, are not readily applicable or inefficient for the case of thermal transients, where the resulting thermal
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Fig. 2. The schematic of an edge-cracked coating resting on a substrate. Fig. 1. The failure mode of a Cr coating/steel substrate exposed to thermal loads.
stress is highly inhomogeneous in the thickness direction. In particular, should the thermal conditions be changed, the thermal stress intensity factor (TSIF) need to be re-calculated even for the same geometric model. The weight function is well known to be a unique feature of geometry, which makes it the right and costeffective means to overcome the above deficiencies [14]. With the weight function for a particular cracked body being known, the TSIFs for any thermal stress distribution can be determined by implementing a simple quadrature procedure. For homogeneous materials, the weight functions have been derived for a wide variety of crack geometries [15–19]. In contrast, the counterpart data for the coated materials seem rather limited in the published literature. To name a few, Fett et al. [20] developed the weight function for an edge crack within the thin surface layer. However, the results were presented only in limited crack lengths without an easy-to-use formula. More recently, Chen and You [21] obtained the weight function for edge cracks located in a segmented coating. The fitted formulas for such important functions were indeed achieved but are not suitable for thermal loading application. The objective of this work is, by the weight function method, to determine the thermally induced TSIF for an edge crack in the coating. Different thicknesses and heat transfer conditions are chosen on the underlying substrate to model different situations in practical application. The transient temperature and thermal stress field for the un-cracked coating-substrate system are first established in closed forms. The weight function applicable to thermal loading is then developed by using a three-parameter approach. Finally, the principle of superposition is adopted and the transient TSIF for an edge crack is evaluated by using the derived weight function and thermal stresses.
2. Model description and assumptions The edge-cracked coating-substrate system under consideration is shown in Fig. 2, in which the coating layer #1 has a thickness h1 and rests upon the substrate layer #2 of thickness h2. An edge crack, of depth a, is oriented perpendicular to the upper surface of the coating. The rectangular coordinate axes are set up with the horizontal coordinate as y and the downward coordinate along the crack face as x (see Fig. 2). The thermal diffusivity, thermal conductivity, coefficient of thermal expansion, Young’s modulus and Poisson’s ratio are denoted by D, k, a, E and l in that order. All quantities pertaining to coating will be marked with subscript 1, whereas those for substrate will carry subscript 2. Both layer materials are considered to be homogeneous, isotropic, and linearly
thermo-elastic. For simplicity, all thermo-elastic coupling effects, inertia effects and possible temperature dependence of thermal and mechanical properties are neglected. 3. Temperature and thermal stresses Suppose the coating–substrate system is initially at a uniform temperature, T0. At time t = 0, the upper surface x = 0 is suddenly exposed to a thermal shock with ambient temperature Ta and is held constant at Ta thereafter. Although actually the temperature can never be instantaneously altered on the coating surface, this treatment can provide an upper bound estimate for most thermal transient problems of practical interest. Let temperature deviations at any time be hi(x, t) = Ti(x, t) T0 and h0 = Ta T0, where Ti(x, t) (i = 1, 2) are the temperatures in the coating and substrate, respectively. In the case of a cold shock, h0 < 0. As the crack plane is in the direction of heat flow, i.e., in the thickness direction, the temperature field will not be altered irrespective of the existence of such a crack. Thus heat does not flow across the crack faces, and instead it flows in the thickness direction. It is noteworthy that a thermalmedium crack model was proposed to simulate the effects of the medium inside cracks [22,23]. The transient temperature field of the coated medium is governed by the heat diffusion equations
@hi ðx; tÞ @ 2 hi ðx; tÞ ¼ Di @t @x2
ði ¼ 1; 2Þ:
ð1Þ
The initial and boundary conditions on the upper surface are given by
hi ðx; 0Þ ¼ 0;
ði ¼ 1; 2Þ
h1 ð0; tÞ ¼ h0 HðtÞ:
ð2aÞ ð2bÞ
H(t) is a unit step function, which is equal to unity for t > 0. In order to examine the effects of different heat transfer schemes applied to the substrate, two thermal conditions on the lower surface are prescribed, which can be stated as Eq. (2c) for case A and as Eq. (2d) for case B (see Fig. 2), respectively:
h2 ðh1 þ h2 ; tÞ ¼ 0
ð2cÞ
@h2 ðh1 þ h2 ; tÞ ¼0 @x
ð2dÞ
The above two cases correspond to two extreme thermal conditions on the lower boundary of the configuration, i.e. constant temperature and perfect insulation. The interface of the coating and substrate is assumed to be perfectly bonded, which means the temperature and heat flux are continuous there, viz.
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h1 ðh1 ; tÞ ¼ h2 ðh1 ; tÞ
ð3aÞ ð3bÞ
By applying the conventional method of Laplace transform [24], one can obtain a closed-form solution of the transient temperature for case A as kð x1Þ h1 ðx;FoÞ ¼ hhþ k h0 1 X 2 g sin b ðx1Þ cos b kh ½cos bm ðx1Þ sin bm kh m m expðFobm Þ ; 2 sin b sin khb ðgþkhÞ cos b cos khb bm ½ð1þgkhÞ m m m m m¼1
ð4aÞ
h2 ðx;FoÞ x ¼ hþ1 k h0 hþ 1 X
2
m¼1
ð4bÞ
xÞexpðFob2m Þ sin kbm ðhþ1 sin b sin khb ðgþkhÞ cos b cos khb : bm ½ð1þgkhÞ m m m m
For case B, we have the temperature field expressed by h1 ðx;FoÞ ¼1 h0 1 X ½cos bm ðx1Þ cos bm khþ 2 g sin b ðx1Þ sin b kh m m expðFobm Þ ; 2 sin b cos khb þðgþkhÞ cos b sin khb bm ½ð1þgkhÞ m m m m m¼1
ð4cÞ
h2 ðx;FoÞ ¼1 h0 1 X expðFob2 Þ bm kðx1Þ sin bm kh ½cos kbm ðx1Þ cos bm khþsin m : 2 sin b cos khb þðgþkhÞ cos b sin khb bm ½ð1þgkhÞ m m m m m¼1
ð4dÞ
1
ues bm in Eqs. (4a) and (4b) are determined from the transcendental Eq. (5a),
þ g sin b cosðkhbÞ ¼ 0: cos b sinðkhbÞ
ð5aÞ
And those in Eqs. (4c) and (4d) are the roots of Eq. (5b),
g sin b sinðkhbÞ ¼ 0: cos b cosðkhbÞ
ð5bÞ
Once the temperature at any time instant is known, the transient thermal stress in the un-cracked and fully free composite plate can be determined in a straightforward manner by using the usual equilibrium equations, thermo-elastic constitutive equations and geometrical equations. Under the assumption of a plane strain condition, the thermal stress component of interest (ryy) has been conveniently written in the following form [10]:
riyy ðx; tÞ ¼
Ei 1 li
e0 ðtÞ þ
x
qðtÞ
ai hi ðx; tÞ
ði ¼ 1; 2Þ:
ð6Þ
The details of deriving Eq. (6) can be found in Ref. [10] and riyy denotes the corresponding value in the coating (i = 1) or in the substrate (i = 2). Since the thermal stress is self-equilibrated, the resultant force and moment at any cross section of the coated media should vanish at any time, viz.
Z
h1 þh2 0
ryy ðx; tÞdx ¼ 0;
Z
wðx; aÞrðxÞdx;
ð8Þ
h1 þh2
ryy ðx; tÞxdx ¼ 0:
where a is the crack depth, r the normal stress along the potential crack in the un-cracked body, and w is the weight function, which can be approximated by [19,20]
# rffiffiffiffiffiffi" 1 X 2 1 j1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ w¼ M ð1 x=aÞ ; pa 1 x=a j¼1 j
ð9Þ
in which the coefficients Mj remain to be determined. The above expression is usually of satisfactory accuracy when truncated after the term j = 3 and thus can be derived using two reference stress intensity factors and an additional condition [19]. Here, one reference loading configuration is chosen as a pair of concentrated forces per thickness, P, loaded at the crack mouth x = 0 and the other is the uniformly distributed stress, r, exerted on the crack faces. The crack tip SIFs for both cases can read, respectively, as
2P K IP ¼ pffiffiffiffiffiffiffiffiffi F P ; ph1
ð10Þ
pffiffiffiffiffiffiffiffiffi K I r ¼ r ph1 F r ;
ð11Þ
where FP and Fr are the dimensionless geometric factors. By applying P-theorem in dimensional analysis [25], the functional relationship of the above geometric factors can be presented in a general form
For convenience, the following dimensionless variables have pffiffiffiffiffiffiffiffiffiffiffiffiffiffi been introduced in Eqs. (4a)–(4d): 1=k ¼ D2 =D1 ; 2 h ¼ h2 =h1 ; ¼ k2 =k1 ; g ¼ kk; x ¼ x=h1 ; Fo ¼ D1 t=h . The eigenvalk
a
0
@ @ h1 ðh1 ; tÞ ¼ k2 h2 ðh1 ; tÞ @x @x
k1
Z K¼
ð7Þ
0
Therefore, the time-dependent parameters, e0(t) and q(t), can be directly determined from Eq. (7) and are listed in Appendix A.
4. Derivation of weight function According to Bueckner’s relation [14], the stress intensity factor, K, for any stress distribution can be expressed as an integral of the product of applied stress and weight function, viz.
F P ¼ P1
a h2 E 2 ; ; ; l1 ; l2 ; h1 h1 E 1
F r ¼ P2
a h2 E2 ; ; ; l1 ; l2 : h1 h1 E1
ð12Þ
ð13Þ
The additional condition is usually of the following form, which indicates that the curvature of the weight function vanishes at the crack mouth [19]:
@2w ¼ 0 for x ¼ 0: @x2
ð14Þ
Upon substitution of Eqs. (9)–(11) into Eqs. (8) and (14), one yields
pffiffiffi qffiffiffiffi 2 ha1 F P 1 qffiffiffiffi pffiffi M1 þ 35 M2 þ 37 M 3 ¼ 3 88p ha1 F r 3 M1 þ M2 þ M3 ¼
ð15Þ
M1 3M 2 15M3 ¼ 3 Then the three M’s can be found by solving the above simultaneous algebraic equations:
qffiffiffiffi pffiffi qffiffiffiffi pffiffi M1 ¼ 158 2 ha1 F P þ 35162p ha1 F r 7 qffiffiffiffi pffiffi qffiffiffiffi pffiffi M2 ¼ 154 2 ha1 F P 35122p ha1 F r þ 25 3 qffiffiffiffi pffiffi qffiffiffiffi pffiffi M3 ¼ 7 8 2 ha1 F P þ 35482p ha1 F r 73
ð16Þ
The Finite Element Method (FEM) has been used to calculate reference SIFs (KIP and KIr) and hence the corresponding geometric factors. The analysis was performed using the commercial package ANSYS 11.0 standard code [26]. The size of the coated medium in ydirection was chosen to be 80 times the coating thickness, which was confirmed to be able to represent the infinite extent in this direction. In order to obtain accurate results, the mesh was sufficiently refined around the crack tip and quarter-point singularity elements have been used to pick up a square-root singularity in the stress at the crack tip. A typical finite element meshing around an edge crack in the coating is shown in Fig. 3.
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
Substrate
Coating
Crack Face
103
Interface
Fig. 3. Typical finite element meshing around an edge crack in the coating.
The reference SIFs are computed by displacement extrapolation method through the following relation [27]:
Table 1 Properties of the used material pair [28]. D2/D1
k2/k1
E2/E1
v1
v2
a2 / a1
0.293
0.389
0.724
0.21
0.31
1.923
i uq KI ¼ E y
rffiffiffiffiffiffiffi
p
8r q
;
Fig. 4. The normalized temperature hi/h0 versus x/h1 for different Fo’s: (a) h2/h1 = 10; (b) h2/h1 = 30.
ð17Þ
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where uqy is the y-directional displacement for the quarter point on the free crack face, r q the distance from quarter point to the crack = E/(1 l2) is the plane strain modulus. If the crack tip tip, and E is within the coating, i = 1, while it is penetrating into the substrate, i = 2. Different ratios a/h1, h2/h1 and material properties E2/E1, l1, l2 can be chosen to compute the two reference SIFs KIP and KIr, through which the dimensionless geometric factors FP and Fr are evaluated by Eqs. (10) and (11). The material pair, which corresponds to a Cr coating deposited onto a structural steel substrate 30CrNi2MoVA [28], is considered in this paper. The dimensionless quantities of the thermo-elastic properties and geometrical parameters are tabulated in Table 1. Based on the finite element data, the empirical formula of FP and Fr by curve-fitting (with an adjusted R-square equal to unity) were obtained and listed in Appendix B. Finally, the coefficients M1, M2 and M3 are calculated by Eq. (16) and the weight functions are thus derived via Eq. (9).
5. Results and discussion 5.1. Temperature and thermal stresses The normalized transient temperature hi(x, t)/h0 versus the normalized depth x/h1 in the coated medium, defined by Eqs. (4a)– (4d), have been plotted in Fig. 4, where the results are shown for two thickness ratios (h2/h1 = 10, 30) and different normalized times (Fourier number). The cases A and B correspond to prescribed constant temperature and perfectly insulated on the lower surface of the substrate, respectively (see Fig. 2). At a constant time instant, the normalized temperature for both cases are seen to decrease as expected with the increase of x/h1. In addition, as the time increases, the normalized temperature ‘‘penetrates” more deeply along the thickness direction, and its gradient decreases as well. At a fixed x/ h1, the normalized temperature increases monotonically until the steady state is approached. On either side of the interface x/h1 = 1, each curve has different slopes, which is due to different thermal resistances of the coating and substrate. For short time intervals,
Fig. 5. The normalized thermal stress ryy/r0 versus x/h1 for different Fo’s: (a) h2/h1 = 10; (b) h2/h1 = 30.
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the two normalized temperature distribution curves for both case A and case B seem to be almost coincident. The associated curves begin to exhibit noticeable difference after the normalized time Fo 100 for h2/h1 = 10 and Fo 500 for h2/h1 = 30, respectively. The variation of the normalized thermal stress component ryy/r0 in the un-cracked medium versus normalized x-coordinate, as obtained by Eq. (6), is presented in Fig. 5 for different normalized time instants. In both graphs, r0 = E1a1h0/(1 l1) and the results are again demonstrated for two values of thickness ratio (h2/h1 = 10, 30). In overview, for short time durations the thermal stresses start to be positive (tensile stresses) in the coating and near-interface substrate, followed by negative (compressive stresses) in the interior substrate, and positive again (tensile stresses) near the lower surface. As can be seen, the thermal stresses in the coating are non-uniform along the thickness direction (x-direction) and its gradient is a decreasing function of time. The maximum value of tensile stress in the coating, rmax = r0, is achieved at the beginning of a cold shock (Fo = 0) and this is the thermal stress in a fully constrained plate undergoing a sudden temperature change. The edge crack is thus expected to initiate at this site when rmax is higher than the strength of coating material.
Table 2 Comparison of geometric factors FP* of our results and those from Ref. [20], which pffiffiffiffiffiffi were calculated by J-integral method. l1 ¼ l2 ¼ 0:3, h2 =h1 ¼ 19, F P ¼ K IP pa=2P. F P and F 0P are results from Ref. [20] and this paper, respectively. E2/E1
a/h1
FP*
F0 P*
Difference (%)
50 50 10 10 3 3 1 1 1/3 1/3 1/10 1/10 1/50 1/50 1/50 1/50
0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5
0.5848 1.9787 0.6708 1.8669 0.8838 1.6574 1.3214 1.3904 2.0911 1.1625 3.3284 1.0414 5.4566 1.0081 5.4566 1.0081
0.5841 1.9770 0.6710 1.8656 0.8855 1.6567 1.3247 1.3901 2.0955 1.1632 3.3357 1.0420 5.4672 1.0086 5.4672 1.0086
0.12 0.09 0.03 0.07 0.19 0.04 0.25 0.02 0.21 0.06 0.22 0.05 0.19 0.05 -0.19 -0.05
Since the coating and substrate have dissimilar thermo-elastic properties, the normalized thermal stress turns out to discontinue
Fig. 6. The normalized TSIF versus normalized time for various normalized crack depths (a/h1 < 1): (a) h2/h1 = 10; (b) h2/h1 = 30.
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X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
Table 3 Comparison of geometric factors Fr of our results and those from Ref. [20], which pffiffiffiffiffiffi were calculated by J-integral method. l1 ¼ l2 ¼ 0:3, h2 =h1 ¼ 19, F r ¼ K Ir =r pa. F r and F 0r are results from Ref. [20] and this paper, respectively.
Table 4 The comparison of some SIFs for an edge-cracked, homogeneous strip, whose crack faces are subjected to uniform pressure r, K I Z is the result of current paper and K 0I from handbook [29]. a is the crack depth, and the strip thickness h = h1 + h2.
E2/E1
a/h1
Fr*
F0 r*
Difference (%)
a /h
pffiffiffiffiffiffi K1/(r pa)
pffiffiffiffiffiffi K0 1/(r pa)
Difference (%)
50 50 10 10 3 3 1 1 1/3 1/3 1/10 1/10 1/50 1/50 1/50
0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 1.5
0.7784 1.4898 0.8207 1.4262 0.9241 1.3087 1.1310 1.1613 1.4754 1.0296 1.9807 0.9425 2.7616 0.9001 0.9001
0.7787 1.4888 0.8217 1.4252 0.9262 1.3087 1.1340 1.1617 1.4782 1.0303 1.9842 0.9437 2.7663 0.9013 0.9013
0.04 0.07 0.12 0.07 0.23 0.00 0.27 0.03 0.19 0.07 0.18 0.13 0.17 0.13 -0.13
1/400 1/300 1/200 1/150 1/100 1/90 1/80 1/70 1/65 1/63 1/61 1/61
1.1210 1.1221 1.1225 1.1222 1.1232 1.1237 1.1241 1.1240 1.1235 1.1232 1.1228 1.1228
1.1215 1.1213 1.1211 1.1209 1.1207 1.1207 1.1207 1.1208 1.1209 1.1209 1.1210 1.1210
0.04 0.06 0.13 0.12 0.22 0.27 0.30 0.29 0.24 0.21 0.16 0.16
at the interface as expected. As time elapses, the tensile thermal stress at the upper surface of the coating becomes smaller in magnitude and finally negative sign takes place since the coefficient of
thermal expansion of Cr coating is almost two times that of steel substrate. Again, for the short time intervals (Fo < 100 for h2/h1 = 10 and Fo < 500 for h2/h1 = 30), there is no obvious difference in the thermal stress between case A and case B, which can be well understood by considering the corresponding temperature distribution in Fig. 4.
Fig. 7. Same as Fig. 6 but Fo < 6: (a) h2/h1 = 10; (b) h2/h1 = 30.
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
5.2. Thermal stress intensity factors The mode I TSIFs associated with the thermal stress have been evaluated by substituting Eqs. (6) and (9) into Eq. (8). To verify the calculations, a series of runs were performed to duplicate selected dimensionless geometric factors FP⁄ and Fr⁄ obtained by Fett et al. [20]. The comparison results are tabulated in Tables 2 and 3, with relative difference less than 0.3%. By letting the coating and substrate have the same material constants, some SIFs for an edgecracked homogeneous strip, derived by the present weight function method, are also found in excellent agreement with the existing data in handbook [29] (see Table 4). Hence the satisfactory accuracy of the weight function method is reasonably validated. The normalized thermal stress intensity factors (TSIFs), KI/K0, versus normalized time, Fo, for an edge crack with selected crack depths ranging from 0.1 to 0.9, are illustrated Figs. 6–8. In these pffiffiffiffiffiffiffiffiffi graphs, K 0 ¼ r0 ph1 : Figs. 6 and 7 correspond to the crack tip within the coating (a/h1 < 1) and the crack tip across the interface (a/h1 > 1) is considered in Fig. 8. Moreover, the TSIF comparisons of the current study with those from finite element solutions are provided in Table 5 for thermal loading conditions as specific val-
107
idation cases. The difference between FEM and current weight function method is within 4%, showing again good agreement. As shown in Fig. 6, the overall variation trends of these curves are qualitatively similar to each other, regardless of the substrate/coating thickness ratios. For any given relative crack depth, the TSIF is observed to initially increase sharply to a local peak value, Kmax, after a finite time and then drops to a plateau as time further increases. If the substrate/coating thickness ratio (h2/h1) is fixed, the TSIF for case A is identical to that for case B during early time intervals despite slight difference at a very later time. It is worthwhile to note that the magnitude of local peak TSIF in such a case is independent of the prescribed thermal conditions on the lower surface of substrate (case A or case B, see Fig. 2). From the point view of linear elastic fracture mechanics, the edge crack will grow when the peak TSIF is high enough to exceed the fracture toughness of the coating. Therefore for the present coating/substrate material pair, the selection of different thermal conditions on the lower surface does not lead to significant difference in the driving force for edge cracking. In order to show more detailed variation pattern of the normalized TSIF during the early time intervals, we re-plotted in Fig. 7 the
Fig. 8. The normalized TSIF versus normalized time for various normalized crack depths (a/h1 > 1): (a) h2/h1 = 10; (b) h2/h1 = 30.
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Table 5 pffiffiffiffiffiffiffiffiffi The comparison of some values of normalized TSIFs, KI/K0, of our results (WF) and those from finite element method (FEM). K 0 ¼ r0 ph1 and r0 = E1a1h0/(1 l1). a/h1 = 0.5 h2/h1 = 10 Case A
Fo WF FEM Diff. (%)
0.1 0.3245 0.3191 1.67
0.2 0.3781 0.3663 3.12
0.3 0.3947 0.3847 2.53
0.4 0.3991 0.3855 3.41
0.6 0.3939 0.3845 2.39
0.8 0.3814 0.3730 2.20
1 0.3664 0.3579 2.32
a/h1 = 0.6 h2/h1 = 30 Case B
Fo WF FEM Diff. (%)
0.2 0.4620 0.4531 1.93
0.4 0.5387 0.5228 2.95
0.6 0.5707 0.5638 1.21
0.8 0.5861 0.5722 2.37
1 0.5936 0.5817 2.00
2 0.5909 0.5813 1.62
4 0.5511 0.5327 3.34
above graphs in a narrower range (Fo = 0–6) of Fourier number. It is clearly shown that, in contrast to the time occurrence (t = 0) of the peak thermal stress (see Fig. 4), the TSIF reaches its local peak value at a delayed time. In addition, the deeper the edge cracks, the later the delayed time. This means that the most severe damage to the edge-cracked medium occurs at an intermediate instant, rather than at the very beginning or at the steady state of a thermal shock. Consequently, the safety evaluations only on such particular time instants might underestimate the severity of a thermal shock. The dependence of local peak TSIF, Kmax, on the relative crack depth, a/h1, can be deduced from Fig. 7, which is a monotonically increasing function of the crack depth. Also can be revealed in Fig. 7 is that the normalized TSIF increases with increasing substrate/coating thickness ratio, h2/h1. For example, the local peak TSIF for a/h1 = 0.9 increases by about 48% as the thickness ratio increases from 10 to 30. As a consequence, the edge crack is more likely to extend for a thicker substrate or a thinner coating, which may be explained by the change in the level of constraint around the crack tip. Fig. 8 shows the normalized thermal stress intensity factors (TSIFs), KI/K0, versus normalized time, Fo, for an edge crack penetrating into the substrate (a/h1 > 1). These results are demonstrated for two values of thickness ratio (h2/h1 = 10, 30) and several values of relative crack depth. On the whole, these curves take on the variation tendencies similar to those for edge cracks within the coating. More specifically for h2/h1 = 10 (see Fig. 8a), as the relative crack depth increases, the local peak TSIF decreases monotonically. But for h2/h1 = 30 (see Fig. 8b), it starts to increase, passes through a global maximum value at a critical crack depth and decreases eventually. This means that an edge crack undergoing thermal shock will be arrested in the substrate or that under thermal shock alone it is not possible for the crack to propagate through the entire thickness of the coated medium. Finally, it is worthwhile to note that, the negative TSIF are predicted in some cases. Physically, a negative TSIF cannot exist as this would mean an interpenetration of crack faces. However, as an intermediate result, it can be considered negative when superimposed on the positive TSIF due to other factors.
coating. These results may be helpful for safety analysis of the coating-substrate system under thermal transients.
6. Conclusions
Appendix B
In this paper, the thermal shock induced cracking behavior of an edge crack in a brittle coating has been investigated, with two thermal conditions considered on the lower surface of the substrate. The weight functions suitable for thermal loading were developed for the edge-cracked coating by using a three-parameter approach. Thermal stress intensity factors (TSIFs) at the crack tip were derived in closed forms. During the short duration of a thermal shock, the time histories of TSIF for both cases were shown to almost coincide for any crack depths, but the two curves begin to exhibit noticeable difference at a certain delayed time and afterwards. The most severe damage to the edge-cracked plate occurs at an intermediate instant, rather than at the very beginning or at the steady state of a thermal shock. The TSIF was also found to depend strongly on the substrate/coating thickness ratio, which becomes higher in magnitude for a thicker substrate or a thinner
Acknowledgments The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51171026). Appendix A
e0 ¼
AC 3 BC 2
1
AC 2 BC 1
q
¼
ðA:1Þ
C 1 C 3 C 22
ðA:2Þ
C 22 C 1 C 3
E1 E2 R1 þ R2 ; 1 l1 1 l2 E1 E2 N1 þ N2 B¼ 1 l1 1 l2 A¼
ðA:3Þ
E1 E2 h1 þ h2 1 l1 1 l2
C1 ¼
ðA:4Þ
C2 ¼
1 E1 1 E2 2 2 2 h1 þ ½ðh1 þ h2 Þ h1 2 1 l1 2 1 l2
ðA:5Þ
C3 ¼
1 E1 1 E2 3 3 3 h1 þ ½ðh1 þ h2 Þ h1 3 1 l1 3 1 l2
ðA:6Þ
Z
h1
R1 ¼
a1 h1 ðx; tÞdx; R2 ¼
Z
0
Z N1 ¼
a2 h2 ðx; tÞdx
ðA:7Þ
h1 h1
a1 h1 ðx; tÞxdx; N2 ¼
0
n¼
h1 þh2
Z
h1 þh2
a2 h2 ðx; tÞxdx
ðA:8Þ
h1
a h1
ðB:1Þ n
n
n
F P ¼ 3:96578e0:04637 þ 2:6606e0:16543 þ 1:61009e1:632 þ 0:69244 n
n
F ¼ 0:25796e0:14674 24103:2257e27570:96095 þ 24103:62459 r a < h1 ; hh21 ¼ 10 ðB:2Þ n
n
n
F P ¼ 4:05349e0:0462 þ 2:67031e0:16695 þ 4:02695e4:13891 1:74672 n 0:18784
F ¼ 0:30424e r a < h1 ; hh21 ¼ 30
15596:1087e
n 20571:61379
þ 15596:56777 ðB:3Þ
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109 n
F P ¼ 1:95142e4:61379 0:62819n 0:48489 n
n
F ¼ 0:10006e1:60782 þ 10:60163e21:43689 10:13807 r a > h1 ; hh21 ¼ 10
ðB:4Þ
n
F P ¼ 1:31196e1:14182 þ 0:01009n þ 0:68208 n n F r ¼ 23:13347e103:49981 23:1324e103:74754 þ 46:95598 a > h1 ; hh21 ¼ 30
ðB:5Þ
References [1] J.W. Hutchinson, Z. Suo, Mixed mode cracking in layered materials, Adv. Appl. Mech. 29 (1992) 63–191. [2] R.A. Miller, Thermal barrier coatings for aircraft engines: history and directions, J. Therm. Spray Technol. 6 (1997) 35–42. [3] J.H. Underwood, A.P. Park, G.N. Vigllante, P.J. Cote, Thermal damage, cracking and rapid erosion of cannon bore coatings, ASME J. Press. Ves. Technol. 125 (2003) 299–304. [4] H. Li, G. Chen, K. Zhang, G. Luo, Z. Ye, Degradation failure features of chromium-plated gun barrels with a laser-discrete-quenched substrate, Surf. Coat. Technol. 201 (2007) 9558–9564. [5] J.L. He, K.C. Chen, A. Davison, Improvements in the understanding and application of duplex coating systems using arc plasma technology, Surf. Coat. Technol. 200 (2005) 1464–1471. [6] Y.Z. Tsai, J.G. Duh, Thermal stability and microstructure characterization of CrN/WN multilayer coatings fabricated by ion-beam assisted deposition, Surf. Coat. Technol. 200 (2005) 1683–1689. [7] J.H. Kim, M.C. Kim, C.G. Park, Evaluation of functionally graded thermal barrier coatings fabricated by detonation gun spray technique, Surf. Coat. Technol. 168 (2003) 275–280. [8] R.G. Saint-Jacques, F. Bordeaux, B. Stansfield, G. Veilleux, W.W. Zuzak, A. Lakhsasi, C. Boucher, Enhanced resistance of plasma-sprayed TiC coatings to thermal shock, J. Nucl. Mater. 191–194 (1992) 465–468. [9] H. Li, G. Chen, G. Zhang, K. Zhang, G. Luo, Characteristics of the interface of a laser-quenched steel substrate and chromium electroplate, Surf. Coat. Technol. 201 (2006) 902–907.
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[10] A.A. Rizk, Stress intensity factor for an edge crack in two bonded dissimilar materials under convective cooling, Theor. Appl. Fract. Mech. 49 (2008) 251– 267. [11] B. Zhou, K. Kokini, Effect of surface pre-crack morphology on the fracture of thermal barrier coatings under thermal shock, Acta Mater. 52 (2004) 4189– 4197. [12] G.W. Schulze, F. Erdogan, Periodic cracking of elastic coatings, Int. J. Solids Struct. 35 (1998) 3615–3634. [13] X. Chen, Q. Liu, Q. Ma, A parametric investigation on thermally driven edge cracking of a coating/substrate system, J. Coat. Technol. Res. 9 (2012) 541–549. [14] H.F. Bueckner, A novel principle for the computation of stress intensity factors, ZAMM 50 (1970) 529–546. [15] H.J. Petroski, J.D. Achenbach, Computation of the weight function from a stress intensity factor, Eng. Fract. Mech. 10 (1978) 257–266. [16] X.R. Wu, Approximate weight functions for center and edge cracks in finite bodies, Eng. Fract. Mech. 20 (1984) 35–49. [17] R. Oliveira, X.R. Wu, Stress intensity factors for axial cracks in hollow cylinders subjected to thermal shock, Eng. Fract. Mech. 27 (1987) 185–197. [18] G. Shen, G. Glinka, Determination of weight functions from reference stress intensity factors, Theor. Appl. Fract. Mech. 15 (1991) 237–245. [19] T. Fett, Direct determination of weight functions from reference loading cases and geometrical conditions, Eng. Fract. Mech. 42 (1992) 435–444. [20] T. Fett, E. Diegele, D. Munz, G. Rizzi, Weight functions for edge cracks in thin surface layers, Int. J. Fract. 81 (1996) 205–215. [21] X. Chen, Y. You, Weight functions for multiple edge cracks in a coating, Eng. Fract. Mech. 116 (2014) 31–40. [22] X.C. Zhong, K.Y. Lee, A thermal-medium crack model, Mech. Mater. 51 (2012) 110–117. [23] X.C. Zhong, B. Wu, K.S. Zhang, Thermally conducting collinear cracks engulfed by thermomechanical field in a material with orthotropy, Theor. Appl. Fract. Mech. 65 (2013) 61–68. [24] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1986. [25] G.I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996. [26] ANSYS Release 11.0, ANSYS Inc., Canonsburg, PA, 2009. [27] P.P. Lynn, A.R. Ingraffea, Transition elements to be used with quadratic point crack-tip elements, Int. J. Numer. Methods Eng. 12 (1978) 1031–1036. [28] Q.F. Ma, R.S. Fang, L.C. Xiang, S. Guo, Practical Handbook of Thermophysical Properties, Chinese Agricultural Press, Beijing, 1986 (in Chinese). [29] H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, ASME Press, NewYork, 2000.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Effects of substrate thickness and heat transfer scheme on edge cracking of a brittle coating due to thermal transients Xuejun Chen a,b,⇑, Xiuping Wang a a b
Department of Applied Mechanics, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, China Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 25 October 2016 Revised 2 March 2017 Accepted 8 March 2017 Available online 9 March 2017 Keywords: Coating Edge cracking Weight function Thermal stress intensity factor (TSIF)
a b s t r a c t Brittle coatings may suffer edge cracking due to thermal transients. The thermal stress intensity factor (TSIF) is used here to characterize the driving force for such damage modes. The purpose of this paper is to investigate the effect of substrate thickness and heat transfer scheme on the edge cracking behavior. The transient temperature and thermal stress field of the un-cracked system are first obtained in closed forms. The weight functions suitable for thermal loading are then developed for the edge-cracked coating. The TSIF at the crack tip is finally obtained based on the principle of superposition, with the equal thermal stresses utilized as the crack surface tractions. The dependence of the normalized TSIF is examined on different thermal boundary condition, normalized time, and relative crack depth as well as substrate/coating thickness ratio. It is found the thicker substrate leads to much higher driving force for the edge cracking. The heat transfer schemes on the lower surface of the substrate, however, have insignificant effect on it. The findings of this study may assist in the integrity analysis of coatings due to rapidly changing thermal environments. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Protective coatings are usually deposited onto underlying substrates in a broad range of technological applications, either to enhance certain mechanical properties of substrate materials or to endow the surface with new properties (e.g. hardness, corrosion and thermal resistance) [1–3]. In most cases, such thin coatings need to bear very high stresses caused, for example, by mechanical loads, temperature gradients or other sources. Consequently, once the in-plane stress reaches a critical condition, edge cracks may be produced near the surface [1,2]. Under repeated loading, these cracks would then result in more serious interfacial delamination and final removal of coating upon further extending toward and near the interface (see Fig. 1). Even if the coating spallation were not induced, edge cracks touching on the interface may still act as convenient tunnels for corrosive species to flow in and damage the substrate and interface. This scenario regularly occurs in corrosive environments, a typical example of which can be found in a Cr coated gun barrel during the firing process [3,4]. Therefore, a major challenge facing the coating-substrate system lies with that the ⇑ Corresponding author at Department of Applied Mechanics, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, China. E-mail addresses: [email protected] (X. Chen), [email protected] (X. Wang). http://dx.doi.org/10.1016/j.tafmec.2017.03.006 0167-8442/Ó 2017 Elsevier Ltd. All rights reserved.
coating shall be able to remain intact and adhered to substrate throughout the service period. One way to retain the integrity of coating is to delay or avoid edge cracking. Hence various concepts of microstructure design were proposed in materials community such as duplex coating, multilayer coating as well as functionally gradient coating [5–7]. The goal for such designs is to alleviate the misfit strain between the coating and substrate and ultimately to reduce the resulting stress level. In parallel efforts, a lot of processing techniques have been recently developed, the intention of which is to toughen coatings or the interface, and thereby to enhance their crack growth resistance [8,9]. From the point view of fracture mechanics, it can also be achieved by reducing the driving force for edge crack initiation and propagation. This requires the stress intensity factor (SIF) of edge cracks be quantified as a function of crack depth, externally applied loads and material constants as well. By knowing what and how main parameters affect the SIFs, one can make a reliable prediction of crack growth rates and also an optimal selection of associated quantities for cracking suppression. In this paper, we focus on an edge cracking of a brittle coating due to thermal transients. A lot of techniques (e.g. method of singular integral equations, finite element method) have been developed to estimate the SIFs for an edge crack in coatings [10–13]. Most of them, however, are not readily applicable or inefficient for the case of thermal transients, where the resulting thermal
101
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
Fig. 2. The schematic of an edge-cracked coating resting on a substrate. Fig. 1. The failure mode of a Cr coating/steel substrate exposed to thermal loads.
stress is highly inhomogeneous in the thickness direction. In particular, should the thermal conditions be changed, the thermal stress intensity factor (TSIF) need to be re-calculated even for the same geometric model. The weight function is well known to be a unique feature of geometry, which makes it the right and costeffective means to overcome the above deficiencies [14]. With the weight function for a particular cracked body being known, the TSIFs for any thermal stress distribution can be determined by implementing a simple quadrature procedure. For homogeneous materials, the weight functions have been derived for a wide variety of crack geometries [15–19]. In contrast, the counterpart data for the coated materials seem rather limited in the published literature. To name a few, Fett et al. [20] developed the weight function for an edge crack within the thin surface layer. However, the results were presented only in limited crack lengths without an easy-to-use formula. More recently, Chen and You [21] obtained the weight function for edge cracks located in a segmented coating. The fitted formulas for such important functions were indeed achieved but are not suitable for thermal loading application. The objective of this work is, by the weight function method, to determine the thermally induced TSIF for an edge crack in the coating. Different thicknesses and heat transfer conditions are chosen on the underlying substrate to model different situations in practical application. The transient temperature and thermal stress field for the un-cracked coating-substrate system are first established in closed forms. The weight function applicable to thermal loading is then developed by using a three-parameter approach. Finally, the principle of superposition is adopted and the transient TSIF for an edge crack is evaluated by using the derived weight function and thermal stresses.
2. Model description and assumptions The edge-cracked coating-substrate system under consideration is shown in Fig. 2, in which the coating layer #1 has a thickness h1 and rests upon the substrate layer #2 of thickness h2. An edge crack, of depth a, is oriented perpendicular to the upper surface of the coating. The rectangular coordinate axes are set up with the horizontal coordinate as y and the downward coordinate along the crack face as x (see Fig. 2). The thermal diffusivity, thermal conductivity, coefficient of thermal expansion, Young’s modulus and Poisson’s ratio are denoted by D, k, a, E and l in that order. All quantities pertaining to coating will be marked with subscript 1, whereas those for substrate will carry subscript 2. Both layer materials are considered to be homogeneous, isotropic, and linearly
thermo-elastic. For simplicity, all thermo-elastic coupling effects, inertia effects and possible temperature dependence of thermal and mechanical properties are neglected. 3. Temperature and thermal stresses Suppose the coating–substrate system is initially at a uniform temperature, T0. At time t = 0, the upper surface x = 0 is suddenly exposed to a thermal shock with ambient temperature Ta and is held constant at Ta thereafter. Although actually the temperature can never be instantaneously altered on the coating surface, this treatment can provide an upper bound estimate for most thermal transient problems of practical interest. Let temperature deviations at any time be hi(x, t) = Ti(x, t) T0 and h0 = Ta T0, where Ti(x, t) (i = 1, 2) are the temperatures in the coating and substrate, respectively. In the case of a cold shock, h0 < 0. As the crack plane is in the direction of heat flow, i.e., in the thickness direction, the temperature field will not be altered irrespective of the existence of such a crack. Thus heat does not flow across the crack faces, and instead it flows in the thickness direction. It is noteworthy that a thermalmedium crack model was proposed to simulate the effects of the medium inside cracks [22,23]. The transient temperature field of the coated medium is governed by the heat diffusion equations
@hi ðx; tÞ @ 2 hi ðx; tÞ ¼ Di @t @x2
ði ¼ 1; 2Þ:
ð1Þ
The initial and boundary conditions on the upper surface are given by
hi ðx; 0Þ ¼ 0;
ði ¼ 1; 2Þ
h1 ð0; tÞ ¼ h0 HðtÞ:
ð2aÞ ð2bÞ
H(t) is a unit step function, which is equal to unity for t > 0. In order to examine the effects of different heat transfer schemes applied to the substrate, two thermal conditions on the lower surface are prescribed, which can be stated as Eq. (2c) for case A and as Eq. (2d) for case B (see Fig. 2), respectively:
h2 ðh1 þ h2 ; tÞ ¼ 0
ð2cÞ
@h2 ðh1 þ h2 ; tÞ ¼0 @x
ð2dÞ
The above two cases correspond to two extreme thermal conditions on the lower boundary of the configuration, i.e. constant temperature and perfect insulation. The interface of the coating and substrate is assumed to be perfectly bonded, which means the temperature and heat flux are continuous there, viz.
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X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
h1 ðh1 ; tÞ ¼ h2 ðh1 ; tÞ
ð3aÞ ð3bÞ
By applying the conventional method of Laplace transform [24], one can obtain a closed-form solution of the transient temperature for case A as kð x1Þ h1 ðx;FoÞ ¼ hhþ k h0 1 X 2 g sin b ðx1Þ cos b kh ½cos bm ðx1Þ sin bm kh m m expðFobm Þ ; 2 sin b sin khb ðgþkhÞ cos b cos khb bm ½ð1þgkhÞ m m m m m¼1
ð4aÞ
h2 ðx;FoÞ x ¼ hþ1 k h0 hþ 1 X
2
m¼1
ð4bÞ
xÞexpðFob2m Þ sin kbm ðhþ1 sin b sin khb ðgþkhÞ cos b cos khb : bm ½ð1þgkhÞ m m m m
For case B, we have the temperature field expressed by h1 ðx;FoÞ ¼1 h0 1 X ½cos bm ðx1Þ cos bm khþ 2 g sin b ðx1Þ sin b kh m m expðFobm Þ ; 2 sin b cos khb þðgþkhÞ cos b sin khb bm ½ð1þgkhÞ m m m m m¼1
ð4cÞ
h2 ðx;FoÞ ¼1 h0 1 X expðFob2 Þ bm kðx1Þ sin bm kh ½cos kbm ðx1Þ cos bm khþsin m : 2 sin b cos khb þðgþkhÞ cos b sin khb bm ½ð1þgkhÞ m m m m m¼1
ð4dÞ
1
ues bm in Eqs. (4a) and (4b) are determined from the transcendental Eq. (5a),
þ g sin b cosðkhbÞ ¼ 0: cos b sinðkhbÞ
ð5aÞ
And those in Eqs. (4c) and (4d) are the roots of Eq. (5b),
g sin b sinðkhbÞ ¼ 0: cos b cosðkhbÞ
ð5bÞ
Once the temperature at any time instant is known, the transient thermal stress in the un-cracked and fully free composite plate can be determined in a straightforward manner by using the usual equilibrium equations, thermo-elastic constitutive equations and geometrical equations. Under the assumption of a plane strain condition, the thermal stress component of interest (ryy) has been conveniently written in the following form [10]:
riyy ðx; tÞ ¼
Ei 1 li
e0 ðtÞ þ
x
qðtÞ
ai hi ðx; tÞ
ði ¼ 1; 2Þ:
ð6Þ
The details of deriving Eq. (6) can be found in Ref. [10] and riyy denotes the corresponding value in the coating (i = 1) or in the substrate (i = 2). Since the thermal stress is self-equilibrated, the resultant force and moment at any cross section of the coated media should vanish at any time, viz.
Z
h1 þh2 0
ryy ðx; tÞdx ¼ 0;
Z
wðx; aÞrðxÞdx;
ð8Þ
h1 þh2
ryy ðx; tÞxdx ¼ 0:
where a is the crack depth, r the normal stress along the potential crack in the un-cracked body, and w is the weight function, which can be approximated by [19,20]
# rffiffiffiffiffiffi" 1 X 2 1 j1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ w¼ M ð1 x=aÞ ; pa 1 x=a j¼1 j
ð9Þ
in which the coefficients Mj remain to be determined. The above expression is usually of satisfactory accuracy when truncated after the term j = 3 and thus can be derived using two reference stress intensity factors and an additional condition [19]. Here, one reference loading configuration is chosen as a pair of concentrated forces per thickness, P, loaded at the crack mouth x = 0 and the other is the uniformly distributed stress, r, exerted on the crack faces. The crack tip SIFs for both cases can read, respectively, as
2P K IP ¼ pffiffiffiffiffiffiffiffiffi F P ; ph1
ð10Þ
pffiffiffiffiffiffiffiffiffi K I r ¼ r ph1 F r ;
ð11Þ
where FP and Fr are the dimensionless geometric factors. By applying P-theorem in dimensional analysis [25], the functional relationship of the above geometric factors can be presented in a general form
For convenience, the following dimensionless variables have pffiffiffiffiffiffiffiffiffiffiffiffiffiffi been introduced in Eqs. (4a)–(4d): 1=k ¼ D2 =D1 ; 2 h ¼ h2 =h1 ; ¼ k2 =k1 ; g ¼ kk; x ¼ x=h1 ; Fo ¼ D1 t=h . The eigenvalk
a
0
@ @ h1 ðh1 ; tÞ ¼ k2 h2 ðh1 ; tÞ @x @x
k1
Z K¼
ð7Þ
0
Therefore, the time-dependent parameters, e0(t) and q(t), can be directly determined from Eq. (7) and are listed in Appendix A.
4. Derivation of weight function According to Bueckner’s relation [14], the stress intensity factor, K, for any stress distribution can be expressed as an integral of the product of applied stress and weight function, viz.
F P ¼ P1
a h2 E 2 ; ; ; l1 ; l2 ; h1 h1 E 1
F r ¼ P2
a h2 E2 ; ; ; l1 ; l2 : h1 h1 E1
ð12Þ
ð13Þ
The additional condition is usually of the following form, which indicates that the curvature of the weight function vanishes at the crack mouth [19]:
@2w ¼ 0 for x ¼ 0: @x2
ð14Þ
Upon substitution of Eqs. (9)–(11) into Eqs. (8) and (14), one yields
pffiffiffi qffiffiffiffi 2 ha1 F P 1 qffiffiffiffi pffiffi M1 þ 35 M2 þ 37 M 3 ¼ 3 88p ha1 F r 3 M1 þ M2 þ M3 ¼
ð15Þ
M1 3M 2 15M3 ¼ 3 Then the three M’s can be found by solving the above simultaneous algebraic equations:
qffiffiffiffi pffiffi qffiffiffiffi pffiffi M1 ¼ 158 2 ha1 F P þ 35162p ha1 F r 7 qffiffiffiffi pffiffi qffiffiffiffi pffiffi M2 ¼ 154 2 ha1 F P 35122p ha1 F r þ 25 3 qffiffiffiffi pffiffi qffiffiffiffi pffiffi M3 ¼ 7 8 2 ha1 F P þ 35482p ha1 F r 73
ð16Þ
The Finite Element Method (FEM) has been used to calculate reference SIFs (KIP and KIr) and hence the corresponding geometric factors. The analysis was performed using the commercial package ANSYS 11.0 standard code [26]. The size of the coated medium in ydirection was chosen to be 80 times the coating thickness, which was confirmed to be able to represent the infinite extent in this direction. In order to obtain accurate results, the mesh was sufficiently refined around the crack tip and quarter-point singularity elements have been used to pick up a square-root singularity in the stress at the crack tip. A typical finite element meshing around an edge crack in the coating is shown in Fig. 3.
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
Substrate
Coating
Crack Face
103
Interface
Fig. 3. Typical finite element meshing around an edge crack in the coating.
The reference SIFs are computed by displacement extrapolation method through the following relation [27]:
Table 1 Properties of the used material pair [28]. D2/D1
k2/k1
E2/E1
v1
v2
a2 / a1
0.293
0.389
0.724
0.21
0.31
1.923
i uq KI ¼ E y
rffiffiffiffiffiffiffi
p
8r q
;
Fig. 4. The normalized temperature hi/h0 versus x/h1 for different Fo’s: (a) h2/h1 = 10; (b) h2/h1 = 30.
ð17Þ
104
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
where uqy is the y-directional displacement for the quarter point on the free crack face, r q the distance from quarter point to the crack = E/(1 l2) is the plane strain modulus. If the crack tip tip, and E is within the coating, i = 1, while it is penetrating into the substrate, i = 2. Different ratios a/h1, h2/h1 and material properties E2/E1, l1, l2 can be chosen to compute the two reference SIFs KIP and KIr, through which the dimensionless geometric factors FP and Fr are evaluated by Eqs. (10) and (11). The material pair, which corresponds to a Cr coating deposited onto a structural steel substrate 30CrNi2MoVA [28], is considered in this paper. The dimensionless quantities of the thermo-elastic properties and geometrical parameters are tabulated in Table 1. Based on the finite element data, the empirical formula of FP and Fr by curve-fitting (with an adjusted R-square equal to unity) were obtained and listed in Appendix B. Finally, the coefficients M1, M2 and M3 are calculated by Eq. (16) and the weight functions are thus derived via Eq. (9).
5. Results and discussion 5.1. Temperature and thermal stresses The normalized transient temperature hi(x, t)/h0 versus the normalized depth x/h1 in the coated medium, defined by Eqs. (4a)– (4d), have been plotted in Fig. 4, where the results are shown for two thickness ratios (h2/h1 = 10, 30) and different normalized times (Fourier number). The cases A and B correspond to prescribed constant temperature and perfectly insulated on the lower surface of the substrate, respectively (see Fig. 2). At a constant time instant, the normalized temperature for both cases are seen to decrease as expected with the increase of x/h1. In addition, as the time increases, the normalized temperature ‘‘penetrates” more deeply along the thickness direction, and its gradient decreases as well. At a fixed x/ h1, the normalized temperature increases monotonically until the steady state is approached. On either side of the interface x/h1 = 1, each curve has different slopes, which is due to different thermal resistances of the coating and substrate. For short time intervals,
Fig. 5. The normalized thermal stress ryy/r0 versus x/h1 for different Fo’s: (a) h2/h1 = 10; (b) h2/h1 = 30.
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X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
the two normalized temperature distribution curves for both case A and case B seem to be almost coincident. The associated curves begin to exhibit noticeable difference after the normalized time Fo 100 for h2/h1 = 10 and Fo 500 for h2/h1 = 30, respectively. The variation of the normalized thermal stress component ryy/r0 in the un-cracked medium versus normalized x-coordinate, as obtained by Eq. (6), is presented in Fig. 5 for different normalized time instants. In both graphs, r0 = E1a1h0/(1 l1) and the results are again demonstrated for two values of thickness ratio (h2/h1 = 10, 30). In overview, for short time durations the thermal stresses start to be positive (tensile stresses) in the coating and near-interface substrate, followed by negative (compressive stresses) in the interior substrate, and positive again (tensile stresses) near the lower surface. As can be seen, the thermal stresses in the coating are non-uniform along the thickness direction (x-direction) and its gradient is a decreasing function of time. The maximum value of tensile stress in the coating, rmax = r0, is achieved at the beginning of a cold shock (Fo = 0) and this is the thermal stress in a fully constrained plate undergoing a sudden temperature change. The edge crack is thus expected to initiate at this site when rmax is higher than the strength of coating material.
Table 2 Comparison of geometric factors FP* of our results and those from Ref. [20], which pffiffiffiffiffiffi were calculated by J-integral method. l1 ¼ l2 ¼ 0:3, h2 =h1 ¼ 19, F P ¼ K IP pa=2P. F P and F 0P are results from Ref. [20] and this paper, respectively. E2/E1
a/h1
FP*
F0 P*
Difference (%)
50 50 10 10 3 3 1 1 1/3 1/3 1/10 1/10 1/50 1/50 1/50 1/50
0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5
0.5848 1.9787 0.6708 1.8669 0.8838 1.6574 1.3214 1.3904 2.0911 1.1625 3.3284 1.0414 5.4566 1.0081 5.4566 1.0081
0.5841 1.9770 0.6710 1.8656 0.8855 1.6567 1.3247 1.3901 2.0955 1.1632 3.3357 1.0420 5.4672 1.0086 5.4672 1.0086
0.12 0.09 0.03 0.07 0.19 0.04 0.25 0.02 0.21 0.06 0.22 0.05 0.19 0.05 -0.19 -0.05
Since the coating and substrate have dissimilar thermo-elastic properties, the normalized thermal stress turns out to discontinue
Fig. 6. The normalized TSIF versus normalized time for various normalized crack depths (a/h1 < 1): (a) h2/h1 = 10; (b) h2/h1 = 30.
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X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
Table 3 Comparison of geometric factors Fr of our results and those from Ref. [20], which pffiffiffiffiffiffi were calculated by J-integral method. l1 ¼ l2 ¼ 0:3, h2 =h1 ¼ 19, F r ¼ K Ir =r pa. F r and F 0r are results from Ref. [20] and this paper, respectively.
Table 4 The comparison of some SIFs for an edge-cracked, homogeneous strip, whose crack faces are subjected to uniform pressure r, K I Z is the result of current paper and K 0I from handbook [29]. a is the crack depth, and the strip thickness h = h1 + h2.
E2/E1
a/h1
Fr*
F0 r*
Difference (%)
a /h
pffiffiffiffiffiffi K1/(r pa)
pffiffiffiffiffiffi K0 1/(r pa)
Difference (%)
50 50 10 10 3 3 1 1 1/3 1/3 1/10 1/10 1/50 1/50 1/50
0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 0.8 1.5 1.5
0.7784 1.4898 0.8207 1.4262 0.9241 1.3087 1.1310 1.1613 1.4754 1.0296 1.9807 0.9425 2.7616 0.9001 0.9001
0.7787 1.4888 0.8217 1.4252 0.9262 1.3087 1.1340 1.1617 1.4782 1.0303 1.9842 0.9437 2.7663 0.9013 0.9013
0.04 0.07 0.12 0.07 0.23 0.00 0.27 0.03 0.19 0.07 0.18 0.13 0.17 0.13 -0.13
1/400 1/300 1/200 1/150 1/100 1/90 1/80 1/70 1/65 1/63 1/61 1/61
1.1210 1.1221 1.1225 1.1222 1.1232 1.1237 1.1241 1.1240 1.1235 1.1232 1.1228 1.1228
1.1215 1.1213 1.1211 1.1209 1.1207 1.1207 1.1207 1.1208 1.1209 1.1209 1.1210 1.1210
0.04 0.06 0.13 0.12 0.22 0.27 0.30 0.29 0.24 0.21 0.16 0.16
at the interface as expected. As time elapses, the tensile thermal stress at the upper surface of the coating becomes smaller in magnitude and finally negative sign takes place since the coefficient of
thermal expansion of Cr coating is almost two times that of steel substrate. Again, for the short time intervals (Fo < 100 for h2/h1 = 10 and Fo < 500 for h2/h1 = 30), there is no obvious difference in the thermal stress between case A and case B, which can be well understood by considering the corresponding temperature distribution in Fig. 4.
Fig. 7. Same as Fig. 6 but Fo < 6: (a) h2/h1 = 10; (b) h2/h1 = 30.
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
5.2. Thermal stress intensity factors The mode I TSIFs associated with the thermal stress have been evaluated by substituting Eqs. (6) and (9) into Eq. (8). To verify the calculations, a series of runs were performed to duplicate selected dimensionless geometric factors FP⁄ and Fr⁄ obtained by Fett et al. [20]. The comparison results are tabulated in Tables 2 and 3, with relative difference less than 0.3%. By letting the coating and substrate have the same material constants, some SIFs for an edgecracked homogeneous strip, derived by the present weight function method, are also found in excellent agreement with the existing data in handbook [29] (see Table 4). Hence the satisfactory accuracy of the weight function method is reasonably validated. The normalized thermal stress intensity factors (TSIFs), KI/K0, versus normalized time, Fo, for an edge crack with selected crack depths ranging from 0.1 to 0.9, are illustrated Figs. 6–8. In these pffiffiffiffiffiffiffiffiffi graphs, K 0 ¼ r0 ph1 : Figs. 6 and 7 correspond to the crack tip within the coating (a/h1 < 1) and the crack tip across the interface (a/h1 > 1) is considered in Fig. 8. Moreover, the TSIF comparisons of the current study with those from finite element solutions are provided in Table 5 for thermal loading conditions as specific val-
107
idation cases. The difference between FEM and current weight function method is within 4%, showing again good agreement. As shown in Fig. 6, the overall variation trends of these curves are qualitatively similar to each other, regardless of the substrate/coating thickness ratios. For any given relative crack depth, the TSIF is observed to initially increase sharply to a local peak value, Kmax, after a finite time and then drops to a plateau as time further increases. If the substrate/coating thickness ratio (h2/h1) is fixed, the TSIF for case A is identical to that for case B during early time intervals despite slight difference at a very later time. It is worthwhile to note that the magnitude of local peak TSIF in such a case is independent of the prescribed thermal conditions on the lower surface of substrate (case A or case B, see Fig. 2). From the point view of linear elastic fracture mechanics, the edge crack will grow when the peak TSIF is high enough to exceed the fracture toughness of the coating. Therefore for the present coating/substrate material pair, the selection of different thermal conditions on the lower surface does not lead to significant difference in the driving force for edge cracking. In order to show more detailed variation pattern of the normalized TSIF during the early time intervals, we re-plotted in Fig. 7 the
Fig. 8. The normalized TSIF versus normalized time for various normalized crack depths (a/h1 > 1): (a) h2/h1 = 10; (b) h2/h1 = 30.
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X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109
Table 5 pffiffiffiffiffiffiffiffiffi The comparison of some values of normalized TSIFs, KI/K0, of our results (WF) and those from finite element method (FEM). K 0 ¼ r0 ph1 and r0 = E1a1h0/(1 l1). a/h1 = 0.5 h2/h1 = 10 Case A
Fo WF FEM Diff. (%)
0.1 0.3245 0.3191 1.67
0.2 0.3781 0.3663 3.12
0.3 0.3947 0.3847 2.53
0.4 0.3991 0.3855 3.41
0.6 0.3939 0.3845 2.39
0.8 0.3814 0.3730 2.20
1 0.3664 0.3579 2.32
a/h1 = 0.6 h2/h1 = 30 Case B
Fo WF FEM Diff. (%)
0.2 0.4620 0.4531 1.93
0.4 0.5387 0.5228 2.95
0.6 0.5707 0.5638 1.21
0.8 0.5861 0.5722 2.37
1 0.5936 0.5817 2.00
2 0.5909 0.5813 1.62
4 0.5511 0.5327 3.34
above graphs in a narrower range (Fo = 0–6) of Fourier number. It is clearly shown that, in contrast to the time occurrence (t = 0) of the peak thermal stress (see Fig. 4), the TSIF reaches its local peak value at a delayed time. In addition, the deeper the edge cracks, the later the delayed time. This means that the most severe damage to the edge-cracked medium occurs at an intermediate instant, rather than at the very beginning or at the steady state of a thermal shock. Consequently, the safety evaluations only on such particular time instants might underestimate the severity of a thermal shock. The dependence of local peak TSIF, Kmax, on the relative crack depth, a/h1, can be deduced from Fig. 7, which is a monotonically increasing function of the crack depth. Also can be revealed in Fig. 7 is that the normalized TSIF increases with increasing substrate/coating thickness ratio, h2/h1. For example, the local peak TSIF for a/h1 = 0.9 increases by about 48% as the thickness ratio increases from 10 to 30. As a consequence, the edge crack is more likely to extend for a thicker substrate or a thinner coating, which may be explained by the change in the level of constraint around the crack tip. Fig. 8 shows the normalized thermal stress intensity factors (TSIFs), KI/K0, versus normalized time, Fo, for an edge crack penetrating into the substrate (a/h1 > 1). These results are demonstrated for two values of thickness ratio (h2/h1 = 10, 30) and several values of relative crack depth. On the whole, these curves take on the variation tendencies similar to those for edge cracks within the coating. More specifically for h2/h1 = 10 (see Fig. 8a), as the relative crack depth increases, the local peak TSIF decreases monotonically. But for h2/h1 = 30 (see Fig. 8b), it starts to increase, passes through a global maximum value at a critical crack depth and decreases eventually. This means that an edge crack undergoing thermal shock will be arrested in the substrate or that under thermal shock alone it is not possible for the crack to propagate through the entire thickness of the coated medium. Finally, it is worthwhile to note that, the negative TSIF are predicted in some cases. Physically, a negative TSIF cannot exist as this would mean an interpenetration of crack faces. However, as an intermediate result, it can be considered negative when superimposed on the positive TSIF due to other factors.
coating. These results may be helpful for safety analysis of the coating-substrate system under thermal transients.
6. Conclusions
Appendix B
In this paper, the thermal shock induced cracking behavior of an edge crack in a brittle coating has been investigated, with two thermal conditions considered on the lower surface of the substrate. The weight functions suitable for thermal loading were developed for the edge-cracked coating by using a three-parameter approach. Thermal stress intensity factors (TSIFs) at the crack tip were derived in closed forms. During the short duration of a thermal shock, the time histories of TSIF for both cases were shown to almost coincide for any crack depths, but the two curves begin to exhibit noticeable difference at a certain delayed time and afterwards. The most severe damage to the edge-cracked plate occurs at an intermediate instant, rather than at the very beginning or at the steady state of a thermal shock. The TSIF was also found to depend strongly on the substrate/coating thickness ratio, which becomes higher in magnitude for a thicker substrate or a thinner
Acknowledgments The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51171026). Appendix A
e0 ¼
AC 3 BC 2
1
AC 2 BC 1
q
¼
ðA:1Þ
C 1 C 3 C 22
ðA:2Þ
C 22 C 1 C 3
E1 E2 R1 þ R2 ; 1 l1 1 l2 E1 E2 N1 þ N2 B¼ 1 l1 1 l2 A¼
ðA:3Þ
E1 E2 h1 þ h2 1 l1 1 l2
C1 ¼
ðA:4Þ
C2 ¼
1 E1 1 E2 2 2 2 h1 þ ½ðh1 þ h2 Þ h1 2 1 l1 2 1 l2
ðA:5Þ
C3 ¼
1 E1 1 E2 3 3 3 h1 þ ½ðh1 þ h2 Þ h1 3 1 l1 3 1 l2
ðA:6Þ
Z
h1
R1 ¼
a1 h1 ðx; tÞdx; R2 ¼
Z
0
Z N1 ¼
a2 h2 ðx; tÞdx
ðA:7Þ
h1 h1
a1 h1 ðx; tÞxdx; N2 ¼
0
n¼
h1 þh2
Z
h1 þh2
a2 h2 ðx; tÞxdx
ðA:8Þ
h1
a h1
ðB:1Þ n
n
n
F P ¼ 3:96578e0:04637 þ 2:6606e0:16543 þ 1:61009e1:632 þ 0:69244 n
n
F ¼ 0:25796e0:14674 24103:2257e27570:96095 þ 24103:62459 r a < h1 ; hh21 ¼ 10 ðB:2Þ n
n
n
F P ¼ 4:05349e0:0462 þ 2:67031e0:16695 þ 4:02695e4:13891 1:74672 n 0:18784
F ¼ 0:30424e r a < h1 ; hh21 ¼ 30
15596:1087e
n 20571:61379
þ 15596:56777 ðB:3Þ
X. Chen, X. Wang / Theoretical and Applied Fracture Mechanics 90 (2017) 100–109 n
F P ¼ 1:95142e4:61379 0:62819n 0:48489 n
n
F ¼ 0:10006e1:60782 þ 10:60163e21:43689 10:13807 r a > h1 ; hh21 ¼ 10
ðB:4Þ
n
F P ¼ 1:31196e1:14182 þ 0:01009n þ 0:68208 n n F r ¼ 23:13347e103:49981 23:1324e103:74754 þ 46:95598 a > h1 ; hh21 ¼ 30
ðB:5Þ
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