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A numerical fracture analysis of indentation into thin hard films on soft substrates
Engineering Fracture Mechanics 70 (2003) 1323–1338 www.elsevier.com/locate/engfracmech
A numerical fracture analysis of indentation into thin hard films on soft substrates K. Sriram, R. Narasimhan *, S.K. Biswas Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India Received 4 December 2001; received in revised form 10 April 2002; accepted 30 April 2002
Abstract In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the mechanics of fracture of the film due to formation of cylindrical or circumferential cracks extending inwards from the film surface. Also, the role of plastic yielding in the substrate on the above mechanics is studied. To this end, the plastic zone development in the substrate and its influence on the load versus indentation depth characteristics and the stress distribution in the film are first examined. Next, the energy release rate J associated with cylindrical cracks is computed. The variation of J with indentation depth and crack length is investigated. The results show that for cracks located near the indenter axis and at small indentation depth, J decreases over a range of crack lengths, which implies stability of crack growth. This regime vanishes as the location of the crack from the axis increases, particularly for a substrate with low yield strength. Finally, a method for combining experimental load versus indentation depth data with simulation results in order to obtain the fracture energy of the film is proposed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thin hard films; Indentation; Finite elements; Fracture; Energy release rate
1. Introduction It is realised today that one of the most effective ways to protect an engineering component operating in erosive, corrosive or high temperature environment under heavy contact traction is to coat it thinly with a hard material like diamond or a ceramic. Hard coatings on soft substrates may be single or multi-layered epitaxial with thickness ranging from a few nm (as in the case of diamond-like carbon coatings used in magnetic recording) to 100 lm. In such situations, the mechanical integrity of the film should be preserved. The stresses resulting from contact tractions on thin films are generally very high. The hardness of the film enables it to resist plastic yielding that is easily induced in the underlying substrate even when the tractions are small. However, under these conditions, different failure modes may operate in the film. The
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Corresponding author. Tel.: +91-80-394-2959; fax: +91-80-360-0648. E-mail address: [email protected] (R. Narasimhan).
0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0013-7944(02)00112-1
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first type of failure is associated with spalling of the film. With increasing traction, cracks which may be radial [1,2] or circumferential [3] extend deeper into the film and reach the film–substrate interface. If substantial yielding occurs in the substrate, the cracks may deflect and propagate along the interface [1]. There is a third possibility wherein the crack may extend into the substrate. The properties which need to be determined in order to design a coated system are the modulus and strength of the film and fracture toughness of both the film as well as the interface. The effectiveness of conventional techniques (such as micro-hardness test) to determine the properties of the film are severely limited due to the sheer smallness of the film thickness. Also, introduction of a submicrometric pre-crack for determination of the fracture toughness of the film is restricted due to limitations in fabrication. The peel test, scratch test and bending test may be conducted on a comparative basis, to yield a measure of adhesion of the film to the substrate [4]. However, its relationship to more fundamental properties such as strength and fracture toughness of the interface is as yet not clear. In this context, it must be mentioned that Wei and Hutchinson [5] have recently examined the mechanics of the peel test, while Volinsky et al. [6] have reviewed thin film adhesion and proposed relationships between film constitutive properties and interfacial toughness. In view of the above problems associated with the dimensional smallness of thin films, depth sensing nanoindentation has become a popular technique in recent times to provide quantification of the mechanical properties of the film and the interface. Thus, determination of the elastic modulus and hardness of a coated sample from nanoindentation load–displacement characteristics is now well established [7], given the area function associated with the indenter tip. However, methods for obtaining the fracture toughness of the film or interface from these test results are not well developed. Since in most structural applications, the film is brittle, this persists as an important obstacle to thin film design. One of the earliest approaches to resolving this problem was due to Marshall and Lawn [8] who gave a semi-empirical formulation for estimating the fracture toughness from data obtained during indentation by a Vickers indenter. However, this analysis is approximate and also does not take cognizance of the fact that crack growth may produce discontinuities or steps in the load–displacement characteristics. Such steps in nanoindentation test records have been observed experimentally [3,9]. A schematic of an experimental load versus indentation depth curve containing a step will be presented later in Section 4.5. The occurrence of these steps in load–displacement curves has been shown to be associated with the formation of circumferential cracks around the indented zone [3,10,11]. Further, these cracks grow along cylindrical surfaces through the thickness of the film [3,10]. Such type of cracking may indeed be a natural occurrence in films having a columnar grain structure with the axes of the elongated grains being perpendicular to the film surface [3,12]. However, the interpretation of experimental data to obtain the fracture toughness of the film has been based on approximate relationships [2] and a posteriori measurement of crack lengths from the coated sample which is a difficult task. It must be noted that emission of dislocations underneath the indenter may be responsible for the discontinuities in the load–displacement characteristics at indentation depths of the order of a few angstroms as discussed, for example, by Tadmor et al. [13]. This gives rise to strain localization and hardness gradients near the film surface. The continuum (fracture mechanics based) viewpoint adopted in this paper would be applicable only to situations where the film thickness and indentation depths are at least an order of magnitude more than the above size scales. The first step towards providing a sound fracture mechanics methodology for analysing indentation test results was taken by Weppelmann and Swain [3]. They conducted a detailed study of the stress fields that develop during spherical indentation of thin hard films deposited on soft substrates. The focus of their work was to investigate the effects of indenter radius to film thickness ratio and plastic yielding in the substrate on the above fields. The weight function method was employed to compute the stress intensity factors associated with cylindrical cracks in the film. However, a systematic investigation of the mechanics of fracture in the film during indentation was not performed in their work. Further, the use of the weight function
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method entails the assumptions of plane strain (whereas, in reality, the domain is axisymmetric), and small geometry changes in the indented zone. Hence, the primary objective of this paper is to investigate the mechanics of fracture in thin hard films on more ductile substrates due to occurrence of cylindrical cracks during spherical indentation. To this end, the stress fields in the film, as well as the energy release rate associated with cylindrical cracks at different radial locations and indentation depths are studied using finite element analyses. An important aim of this work is to assess the role of plastic yielding in the substrate on the above results. Hence, the J2 flow theory of plasticity is employed to describe the constitutive response of the substrate and different yield strengths are considered. The film is assumed to be isotropic and linear elastic and to be free of residual stresses. The energy release rate is computed from the finite element results continuously throughout the indentation analysis of films with pre-existing cylindrical cracks using the numerically accurate domain integral formulation [14]. Finally, a method is proposed by which finite element simulations can be employed to interpret indentation test results and yield information related to the fracture behaviour of the film.
2. Numerical procedure In this work, axisymmetric finite element analyses of spherical indentation of the film–substrate system is carried out. In the analyses reported in Section 3, the film is assumed as uncracked, whereas, cylindrical cracks extending inwards from the film surface around the zone of indentation are modelled in Section 4. An updated Lagrangian finite element formulation [15] is employed in all these analyses. A schematic of the axisymmetric geometry considered in the simulations undertaken in Section 3 is shown in Fig. 1. The ratio of the film thickness to indenter radius, t=Ri , is chosen as 1.5. The focus of the present work is on brittle cracking of the film and not on debonding of the film–substrate interface. Hence, the film is assumed to be perfectly bonded to the interface. The ratios of the radial dimension of the
Fig. 1. Schematic showing the film–substrate system along with the spherical indenter in the r–z plane.
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film–substrate system and the substrate thickness to the indenter radius, Rs =Ri and ts =Ri , are taken as 300 and 150, respectively, so that boundary effects do not influence the stress distribution near the zone of indentation. The contact between the (rigid) indenter and the film is assumed as frictionless and is modelled using a slideline approach along with a penalty formulation [16]. The film is assumed to be linear elastic, whereas a finite deformation, J2 flow theory of plasticity model [15] is employed for the substrate. The material properties are chosen to be representative of a TiN film on a steel substrate (as in the experiments reported in [3]). Thus, the ratio of the Young’s modulus of the substrate to that of the film, Es =Ef , is taken as 0.4. The Poisson’s ratios of the film and substrate, mf and ms , are assumed as 0.2 and 0.3, respectively. Further, the ratio of the tangent modulus to Young’s modulus, Ets =Es , of the substrate is chosen as 0.04. In order to study the effect of plastic yielding in the substrate on the indentation mechanics, three values of initial yield strain of the substrate, rys =Es ¼ 0:001, 0.003 and 0.009 are considered. The finite element meshes used in the analyses are comprised of four-noded isoparametric quadrilateral elements and are well refined near the zone of indentation. It must be noted that since a continuum analysis with homogeneous material properties is performed here, the results obtained would depend only on the ratios of different length dimensions and not on their absolute values. Thus, for example, from analytical solutions for contact between a sphere and an elastic half-space [17], as well as from dimensional considerations, it may be expected that the load P during indentation of a film bonded to a soft substrate (as in Fig. 1) would have the following functional form, h t Es rys P ¼ Ef R2i f ; ; ; ; other material constants : ð1Þ t Ri Ef Es Here, h=t is the ratio of the indentation depth to film thickness and is a measure of the compressive axial strain in the film at the indenter axis. It should be mentioned that additional length scale considerations may be important at submicron thicknesses where localized strain gradients in the substrate may become dominant.
3. Stress analysis of uncracked film–substrate system It is important to first understand the development of plastic yielding in the substrate during indentation and its influence on the load versus indentation depth characteristics and stress variations in the film. This would enable interpretation of the trends in variation of energy release rate with indentation depth and crack length in the fracture analysis to be performed later. Hence, a detailed analysis of indentation of an uncracked film–substrate system is undertaken in this section. 3.1. Load versus indentation depth The variation of normalized load P =ðEf R2i Þ with indentation depth h=t up to a maximum indentation depth of hm =t ¼ 0:05 and 0.2 are presented in Fig. 2(a) and (b), respectively. In these figures, the solid lines correspond to an initial yield strain of the substrate of rys =Es ¼ 0:009 and the dashed lines to rys =Es ¼ 0:001. Both the loading and unloading segments (starting from maximum indentation depth) are shown in these figures. It can be observed from Fig. 2(a) that for rys =Es ¼ 0:009, the loading and unloading curves coincide with each other, indicating a reversible (elastic) response of the film–substrate system. However, for rys =Es ¼ 0:001, the unloading curve differs from the loading curve even at hm =t ¼ 0:05 (see Fig. 2(a)), which is attributed to the development of plastic yielding in the substrate. As the normalized indentation depth is increased, it may be noticed from Fig. 2(b) that the unloading segments for both cases deviate substantially from the loading curves.
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Fig. 2. Normalized load versus indentation depth for rys =Es ¼ 0:001 and 0.009 corresponding to maximum indentation depth of (a) hm =t ¼ 0:05 and (b) hm =t ¼ 0:2. The unloading curves from maximum indentation depth are also shown.
It can be observed from Fig. 2(a) and (b) that the peak load at a given hm =t increases with the yield strength of the substrate. Also, the P versus h curve becomes almost linear at large h=t (see Fig. 2(b)). In s order to characterize this effect, variations of the type P =ðEf R2i Þ ¼ Aðh=tÞ , where A is a constant, were fitted to the loading curves. The values of the index s corresponding to different ranges of indentation depth are summarized in Table 1 for the three values of rys =Es . It must first be noted that the index s for spherical Table 1 Values of exponent s obtained by fitting curves of the type P =ðEf R2i Þ ¼ Aðh=tÞs to the numerical results corresponding to different rys =Es and range of h=t Range of h=t
rys =Es 0.001
0.003
0.009
0–0.05 0.05–0.1 0.1–0.2
1.4 1.2 1.0
1.46 1.29 1.13
1.48 1.4 1.26
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indentation of an elastic film bonded to a rigid substrate should be 1.5, whereas that pertaining to indentation of an elastic–perfectly plastic bulk solid under large scale yielding is 1 [17]. Now, from Table 1 it may be seen that at low h=t, s is close to 1.5 for all rys =Es which implies linear elastic Hertzian behaviour of the film–substrate system. As h=t increases, the value of s decreases to 1, which is an outcome of plastic yielding in the substrate. This is inferred from the fact that the index s approaches 1 more readily (i.e., at smaller indentation depths) when the substrate yield strength is less and there is considerable substrate plasticity (see Section 3.2).
3.2. Plastic strain contours Rt 1=2 The contours of equivalent plastic strain, p ¼ 0 ð2=3Dpij Dpij Þ dt, where Dpij is the plastic part of the rate of deformation [15], in the substrate at hm =t ¼ 0:2 are shown in normalized coordinates in Fig. 3. This figure pertains to the case rys =Es ¼ 0:003. It can be seen from this figure that the plastic strain contours extend more in the z direction. By contrast, at small h=t, it was found that they spread more in the radial direction. With further indentation, they become circular in shape and finally extend more in the z direction at large h=t (as in Fig. 3). Also, the level of plastic strain at a given distance from the point of intersection of the axis and the interface (i.e., r ¼ 0, z=t ¼ 1) increases with indentation depth. As discussed above, the elastic–plastic boundary in the substrate is not spherically symmetric as assumed in Johnson’s cavity model [17] for indentation of an elastic–perfectly plastic half-space. Hence, the normalized maximum extent of the substrate plastic zone from the point of intersection of the axis and the interface, rp =t, is plotted as a function of normalized indentation depth in Fig. 4. Results pertaining to three values of rys =Es are shown in this figure. It may be seen from this figure that the substrate plastic zone grows strongly with indentation depth, particularly for low rys =Es . Further, at any given indentation depth, the enhancement in rp =t with reduction in rys =Es is quite dramatic. This is not too surprising since the spherical cavity model of Johnson [17] also predicts a strong increase in the plastic zone size in an elastic–perfectly plastic bulk solid with decrease in yield strength.
Fig. 3. Contours of equivalent plastic strain in the substrate for the case rys =Es ¼ 0:003 at h=t ¼ 0:2.
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Fig. 4. Normalized substrate plastic zone size versus indentation depth for rys =Es ¼ 0:001, 0.003 and 0.009.
3.3. Stress distribution in the film The variation of normalized radial stress rrr =Ef along the film surface (i.e., z ¼ 0 in the undeformed configuration) corresponding to h=t ¼ 0:2 is shown in Fig. 5. Results pertaining to rys =Es ¼ 0:001 and 0.009 are presented in this figure. Here, the radial distance r is normalized by the contact radius a. The radial stress distributions displayed in Fig. 5 are typically Hertzian [17] and show a tensile peak just outside the contact zone. The magnitude of the tensile peak decreases with reduction in rys =Es (particularly at large h=t as in Fig. 5). The tensile peak implies that a cylindrical crack located just outside the contact zone and extending inwards (i.e., in the negative z direction) from the film surface is likely to be triggered (see also [3]). Hence, in the fracture analysis conducted later, the energy release rate associated with such cylindrical cracks at different indentation depths is studied.
Fig. 5. Normalized radial stress distribution along the film surface for rys =Es ¼ 0:001 and 0.009 at h=t ¼ 0:2.
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Fig. 6. Variation of normalized stresses through the film thickness for two values of rys =Es and r=a, where a is the contact radius: (a) rrr =Ef versus jzj=t at h=t ¼ 0:05, (b) srz =Ef versus jzj=t at h=t ¼ 0:05, (c) rrr =Ef versus jzj=t at h=t ¼ 0:2 and (d) srz =Ef versus jzj=t at h=t ¼ 0:2.
The variations of rrr =Ef and srz =Ef at two locations outside the contact zone are shown in Fig. 6(a) and (b) as a function of subsurface depth jzj normalized by t, corresponding to h=t ¼ 0:05. Similar stress distributions at h=t ¼ 0:2 are presented in Fig. 6(c) and (d). It can be seen from Fig. 6(a) and (c) that rrr =Ef decreases rapidly from the tensile value at the film surface and becomes compressive at a small distance jzj=t below the surface. It attains a peak compressive value between jzj=t ¼ 0:05 and 0.2 depending upon the radial location and indentation depth. The radial stress is small in magnitude for jzj=t > 0:4 at low indentation depth (see Fig. 6(a)). On the other hand, it may be noticed from Fig. 6(c) that at large indentation depth, rrr =Ef becomes tensile for jzj=t > 0:4. In fact, a strongly tensile radial stress prevails at the film–substrate interface. The tensile radial stress at the interface is higher for lower rys =Es , which along with the observation that it is pronounced at large h=t suggests that it is caused by plastic yielding in the substrate. It may be observed from Fig. 6(b) and (d) that for r=a > 1, srz =Ef increases as jzj=t increases, reaches a peak value between 0.05 to 0.3 depending upon r=a and h=t, and then decreases. At small distances jzj=t below the film surface, srz =Ef reduces strongly with increase in radial distance r=a, whereas at large jzj=t (greater than 0.4), it remains fairly constant as r=a changes from 1.1 to 1.5. Also, srz =Ef tends to zero as jzj=t approaches 1, irrespective of rys =Es and h=t, at least up to h=t ¼ 0:2. On comparing Fig. 6(b) and (d), it can be seen that as h=t increases, the magnitude of the peak srz =Ef increases for both values of r=a, and the location of the peak shifts to higher jzj=t. Further, Fig. 6(d) shows that at h=t ¼ 0:2, the magnitude of srz =Ef for a given r=a decreases substantially with reduction in rys =Es for the entire range of jzj=t from 0 to 1. Thus, while reduction in rys =Es elevates the tensile radial stress at the interface, it strongly reduces the shear stress at any distance jzj=t below the film surface. These trends affect the energy release rate pertaining to cylindrical cracks in the film as will be seen in Section 4.
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4. Fracture analysis The results of the stress analysis conducted in the previous section show that a peak tensile radial stress occurs on the film surface just outside the edge of the contact zone (Fig. 5). Further, as noted in the introduction, experimental results (see, for example, [3]) suggest that circumferential cracks will propagate along vertical surfaces during indentation of the film with spherically tipped indenters. Hence, the mechanics of fracture of the film due to the formation of such cracks is examined in this section.
4.1. Modelling aspects A schematic of the problem addressed in this section is illustrated in Fig. 7. The ratios of the film thickness and other geometrical dimensions to the indenter radius are taken as mentioned in Section 2. Further, as shown in Fig. 7, a pre-existing cylindrical crack of length c located at a radius R is modelled. The values of R=Ri that are chosen in this study are 0.27, 0.36 and 0.46. The maximum normalized indentation depths hm =t up to which the analyses are conducted for these three cases are 0.05, 0.1 and 0.2, respectively. It was found from the results obtained in Section 3 that the normalized contact radii a=Ri for an uncracked film at these indentation depths are slightly less than the three R=Ri values mentioned above. Also, it must be noted that the radius of the contact zone adjacent to the indenter axis at a given indentation depth will decrease due to the presence of a circumferential crack owing to the increase in the compliance of the film. In each of the above cases, fracture analysis is conducted for a range of crack length to film
Fig. 7. Schematic showing in the r–z plane a film with a cylindrical crack of length c, located at radius R, which is bonded to a soft substrate and subjected to spherical nanoindentation.
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thickness ratios, c=t, varying from 0.1 to 0.9. The constitutive response and properties of the film and substrate are chosen as indicated in Section 2. The compressive radial stress prevailing below the film surface (see Fig. 6) leads to crack closure for shallow cracks (irrespective of indentation depth), as well as for deep cracks at small indentation depth. In these cases, the contact between the crack faces is assumed to be frictionless and the associated constraints are imposed by the penalty approach [16]. Thus, shallow cylindrical cracks in the film are essentially subjected to mode II loading. On the other hand, there is opening of the crack faces due to tensile radial stress for deep cracks at large indentation depth and, hence, a mixed-mode state prevails near the tip under these conditions. 4.2. Variation of energy release rate with indentation depth The energy release rate J is computed using the domain integral method [14] from several rectangular domains of dimensions varying from 0:01 t 0:01 t to 0:125 t 0:125 t about the crack tip. It is found that J computed from various domains differ by less than 3% for all cases. In order to check the accuracy of these J values, the energy release rate is also obtained for a few cases using the load versus indentation depth curves pertaining to incrementally different crack lengths. In this context, it should be noted that for a cylindrical crack located at radius R from the axis of symmetry in a nonlinear elastic solid, the energy release rate can be defined in terms of the load P versus load point displacement D as follows [18]: ! Z D ~ ; cÞ 1 oP ðD ~: J ¼ dD ð2Þ 2pR 0 oc On applying this relation to the indentation problem, and interpreting D as indentation depth h, it may be argued that the hatched area dA shown in Fig. 8 between P versus h curves corresponding to cracks of
Fig. 8. Schematic of load versus indentation depth curves corresponding to cracks of length c and c þ dc. The shaded area gives J as indicated in the inset.
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Fig. 9. Comparison between normalized J versus h=t variations obtained from P–h curves and the domain integral method for a crack of length c=t ¼ 0:225 located at R=Ri ¼ 0:27, corresponding to rys =Es ¼ 0:003.
length c and c þ dc is given by dA ¼ 2pRJ dc. It must be emphasized that in defining J via Eq. (2), it is assumed that the material behaves as a nonlinear elastic solid. Thus, its application to elastic–plastic materials should be limited to situations where there is monotonic proportional loading, so that the constitutive response can be accurately represented by the deformation theory of plasticity (which is like a nonlinear elasticity model) [18]. In Fig. 9, the normalized J computed from the P versus h variations corresponding to cracks of slightly different lengths (by about 3%) is compared with that obtained from the domain integral method for a typical case. This figure pertains to R=Ri ¼ 0:27 and c=t ¼ 0:225. It may be seen from this figure that the energy release rates computed by the two methods agree closely with each other. Similar agreement was observed for cracks located at larger values of R=Ri and at higher indentation depths. This lends confidence to the energy release rate values obtained from the finite element simulations in this work. In Section 4.5, a method for obtaining the fracture energy Jf of brittle films is suggested by applying Eq. (2) to experimental P versus h records containing steps (which correspond to growth of cylindrical cracks). The variations of normalized energy release rate with indentation depth for cracks located at R=Ri ¼ 0:36 and corresponding to different normalized lengths c=t are shown in Fig. 10. This figure pertains to rys =Es ¼ 0:003. It was found from an analysis of the curves presented in Fig. 10 that J increases approximately as h2:5 for small h=t (up to 0.05), whereas it increases as h2 at large h=t. This change in trend is an outcome of the influence of substrate plasticity on the variation of load with indentation depth which was noted earlier. Thus, the overall fracture response of the film–substrate system is elastic at small h=t, whereas, it is governed by plastic yielding in the substrate at large h=t. Further, it was found that for shallow cracks (say, c=t < 0:4), J at a given h decreases dramatically as the radial location R=Ri of the crack surface increases. By contrast, J is quite independent of R=Ri , at least up to R=Ri ¼ 0:5, for deep cracks at a given h=t. The above trends can be rationalized from Fig. 6(b) and (d) which show a sharp decrease in srz with r=a at small jzj=t, whereas srz is fairly independent of r=a at large jzj=t. It follows that among a series of concentric cracks, the one which is closest to the symmetry axis will be triggered first if the crack length is small. On the other hand, all cracks located over a range of radial distances are likely to be triggered at around the same indentation depth when the crack length is large.
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Fig. 10. Normalized J versus h=t curves corresponding to different crack lengths for cracks located at R=Ri ¼ 0:36. The substrate yield strain is rys =Es ¼ 0:003.
4.3. Variation of energy release rate with crack length It can be observed from Fig. 10 that the energy release rate does not increase monotonically with crack length. In order to understand this effect clearly, the variation of normalized J with crack length c=t is shown in Fig. 11(a)–(c) for indentation depths of h=t ¼ 0:05, 0.1 and 0.2, respectively. These figures pertain to R=Ri ¼ 0:27, 0.36 and 0.46, respectively. In Fig. 11(a) and (b), only results corresponding to rys =Es ¼ 0:003 are presented, whereas in Fig. 11(c) those pertaining to three values of rys =Es are shown. On examining Fig. 11(a), it may be seen that for cracks located close to the symmetry axis and at small indentation depth (h=t ¼ 0:05), J =ðEf Ri Þ increases initially with crack length and reaches a peak at around c=t ¼ 0:2. It then decreases between c=t ¼ 0:2 and 0.75, and thereafter shows a marginal increase. The decreasing branch in the J versus c=t variation is caused by the reduction in the shear stress in the film for jzj=t > 0:2 as seen in Fig. 6(b). It can be observed from Fig. 11(b) that with increase in R=Ri and h=t, the range of crack lengths over which the reduction in J =ðEf Ri Þ occurs, as well as the magnitude of this drop relative to the peak value, are diminished. Finally, it can be seen from the curve pertaining to rys =Es ¼ 0:003 in Fig. 11(c) that at large R=Ri and h=t, the decreasing branch completely vanishes, although J remains almost constant over the range of c=t from 0.3 to 0.6. This is because, at large indentation depth, rrr becomes strongly tensile as the interface is approached (see Fig. 6(c)) which compensates for the reduction in srz with jzj=t. From Fig. 11(c), it can be observed that for any given crack length, the value of J is higher when rys =Es is larger. This counter-intuitive result may be rationalized as follows. First, it must be noted that yielding occurs in the substrate and not near the crack tip in this analysis. Further, it was found that the plastic work dissipated in the substrate is negligible at small indentation depth (h=t < 0:1) compared to the elastic energy of the film, whereas it is reasonably independent of rys =Es at large h=t. On the other hand, the elastic energy of the film increases with rys =Es which becomes pronounced at large h=t. Hence, the influence of rys =Es on J as seen in Fig. 11(c) is an outcome of its effect on the stress distribution in the film. It must be recalled from Fig. 6(d) that at h=t ¼ 0:2, the magnitude of srz is significantly higher for rys =Es ¼ 0:009, irrespective of jzj=t, which results in higher J for this case. Finally, it is interesting to note from Fig. 11(c) that for the case of rys =Es ¼ 0:009, J versus c=t shows a marginally decreasing branch between c=t ¼ 0:3 and 0.6 even at h=t ¼ 0:2.
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Fig. 11. Normalized J versus c=t for cracks (a) located at R=Ri ¼ 0:27 corresponding to h=t ¼ 0:05, (b) located at R=Ri ¼ 0:36 corresponding to h=t ¼ 0:1 and (c) located at R=Ri ¼ 0:46 corresponding to h=t ¼ 0:2.
4.4. Stability of crack growth The decreasing branch of the J versus c=t curves at low h=t for cracks located at small radial distances implies that crack growth over the corresponding range of crack lengths will be stable when the indentation test is carried out under displacement controlled conditions. In other words, the indentation depth will have to increase gradually in order to sustain further crack growth. From Fig. 11(c), it may be seen that the stable branch in the J versus c=t curve persists even for large R=Ri and h=t, if rys =Es is large. However, for low to intermediate strength substrates, growth of cracks located far away from the axis is likely to be unstable at large h=t over the entire range of c=t from 0 to 1 even under displacement controlled conditions since ðoJ =ocÞh > 0 (see Fig. 11(c)). The crack growth behaviour discussed above can also be understood by plotting the normalized P versus h=t curves corresponding to different crack lengths along with lines of constant J =ðEf Ri Þ as shown in Fig. 12. This figure pertains to rys =Es ¼ 0:003 and R=Ri ¼ 0:27. The lines of constant J =ðEf Ri Þ are obtained by connecting combinations of P and h pertaining to various c=t for which J =ðEf Ri Þ is equal to a chosen value. Fig. 12 shows that between c=t ¼ 0:1 and 0.2, the constant J trajectories correspond to decrease in both P and h which results in unstable crack growth under both load and displacement controlled conditions. In the range of c=t between 0.2 to 0.75, crack growth at constant J occurs with increasing h which gives rise to stable crack growth when displacement is prescribed in the indentation test. Finally, for c=t > 0.75, the constant J lines again correspond to decrease in both P and h which leads to unstable crack growth.
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Fig. 12. Normalized P versus h=t curves corresponding to different crack lengths along with constant normalized J trajectories for the case R=Ri ¼ 0:27. The substrate yield strain is rys =Es ¼ 0:003.
4.5. Interpretation of experimental P versus h data As mentioned earlier, experimental data of load versus indentation depth for thin films sometimes display steps which are associated with the formation of cracks in the indented region. These steps could be in the form of an increase in h at constant P or a sudden decrease in the slope of the P versus h curve. In this subsection, a method for interpreting such P versus h data and to obtain meaningful estimates of the fracture energy Jf of the film is presented. In Fig. 13, a schematic of an experimental P versus h curve with a step AB, which corresponds to the growth of a cylindrical crack at a radius R (similar to that observed by Weppelmann and Swain [3]), is shown. In order to obtain Jf from this experimental curve, the extent of crack growth Dc must be first determined. To this end, a family of P versus h curves corresponding to different crack lengths c1 , c2 , etc., at the observed radial location R (similar to Fig. 12), generated by finite element analyses, are superimposed on the experimental data as shown in Fig. 13. In this diagram, point A (beginning of the step in the experimental data) lies between the numerically generated P–h curves pertaining to crack lengths c1 and c2 (and closer to the latter). Similarly, point B may be observed to be close to the curve corresponding to crack length c3 . Hence, the extent of crack growth Dc associated with the step AB may be deduced to be around (c3 c2 ). The fracture energy Jf of the film may now be estimated in two ways. In the first method, Jf may be calculated as Af =ð2pRDcÞ, where Af is the area of the curved wedge OAB in Fig. 13. This approach is based on the application of Eq. (2) and assumes that the substrate material response is like a nonlinear elastic solid (see discussion in Section 4.2). The above assumption will be violated if there is significant elastic unloading in the substrate during crack growth by Dc in the film. In this context, it must be mentioned that this method can also be employed by considering a smaller curved wedge involving points O, A and a point D lying between A and B which is closer to A. Alternatively, Jf may be deduced by conducting further analysis of the results and plotting constant J lines corresponding to levels J1 , J2 , etc., along with the P versus h curves (similar to Fig. 12). Such constant J lines are shown by dash-dot curves in Fig. 13. It may be seen from this figure that the step AB in the experimental data lies close to the constant J line pertaining to the level J2 . Hence, Jf may estimated to be
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Fig. 13. Schematic of experimental load versus displacement data for a circumferentially cracked film (similar to that obtained in Ref. [3]), superposed on a computationally generated nomogram consisting of P versus h curves for different crack lengths and constant J trajectories.
equal to J2 . In fact, with more detailed analysis using the P versus h curves corresponding to different crack lengths and the constant J lines, the variation of Jf with crack length (due to say, small scale yielding in the film near the crack tip) may be deduced. 5. Conclusions The main conclusions of this work are listed below. 1. The load P increases as h3=2 and the unloading segment almost coincides with the loading curve for low indentation depth. This implies linear elastic behaviour of the film–substrate system. As the indentation depth increases, the exponent s characterizing the P versus h variation decreases and tends to unity. Also, the residual depth at complete unloading increases progressively. These trends are attributed to plastic zone development in the substrate. 2. The radial stress is tensile at the film surface outside the contact zone. It becomes compressive at a small distance jzj=t below the film surface. However, at large subsurface depth (jzj=t > 0:4), rrr once again becomes tensile. The tensile radial stress near the film–substrate interface (jzj=t ! 1) increases with h=t and is also higher for lower rys =Es at a given h=t. 3. The shear stress srz increases with distance jzj=t below the film surface, reaches a peak value between 0.05 and 0.3 (depending on the radial location), and then decreases. At large indentation depth, srz at a given distance below the film surface decreases substantially with reduction in rys =Es . 4. There is closure of crack faces for shallow cracks (irrespective of indentation depth) and for deep cracks at small indentation depth. Thus, under these conditions, the cracks are essentially subjected to mode II loading. On the other hand, the crack faces open out for deep cracks at large indentation depth leading to a mixed-mode state.
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5. For cracks located at small R=Ri and at small indentation depth, stable crack growth under displacement controlled conditions is expected to occur between c=t ¼ 0:2 and 0.75, irrespective of rys =Es . This stable crack growth regime vanishes as both R=Ri and h=t increase, albeit at a slower rate for a substrate with larger yield strength. 6. In order to interpret experimental P versus h data which show steps owing to formation of cylindrical cracks, it is suggested that this data be superposed on a computationally generated nomogram consisting of P–h curves pertaining to different crack lengths and constant J lines. This would enable determination of the extent Dc of crack growth, as well as the fracture energy Jf of the film.
References [1] Shiwa M, Weppelman ER, Bendeli A, Swain MV, Munz D, Kishi T. Acoustic emission and precision force–displacement observation of spherical nanoindentation into TiN films on silicon. Surf Coat Technol 1994;68/69:598–602. [2] Haanappel VAC, Vendel DVD, Metselaar HSC, Van Corbach HD, Fransen T, Gellings PJ. The mechanical properties of thin alumina films deposited by metal-organic chemical vapour deposition. Thin Solid Films 1995;254:153–65. [3] Weppelmann E, Swain MV. Investigation of stresses and stress intensity factors responsible for fracture of thin protective coatings during ultra-micro indentation tests with spherical indenters. Thin Solid Films 1996;286:111–21. [4] Kuper A, Clissold R, Martin PJ, Swain MV. A comparative assessment of three approaches for ranking the adhesion of TiN coatings onto two steels. Thin Solid Films 1997;308:329–33. [5] Wei Y, Hutchinson JW. Interface strength, work of adhesion and plasticity in the peel test. Int J Fract 1998;93:315–33. [6] Volinsky AA, Moody NR, Gerberich WW. Interfacial toughness measurements for thin films on substrates. Acta Mater 2002;50:441–66. [7] Pharr GM, Oliver WC. Measurement of thin film mechanical properties using nanoindentation. MRS Bull 1992:28–33. [8] Marshall DB, Lawn BR. Indentation of brittle materials. In: Microindentation Techniques in Materials Science and Engineering. ASTM STP 889. 1984. [9] Whitehead AJ, Page TF. Nanoindentation studies of thin film coated systems. Thin Solid Films 1992;220:277–83. [10] Hainsworth SV, McGurk MR, Page TF. The effect of coating cracking on the indentation response of thin hard-coated systems. Surf Coat Technol 1998;102:97–107. [11] Ding J, Meng Y, Wen S. Mechanical properties and fracture toughness of multilayer hard coatings using nanoindentation. Thin Solid Films 2000;371:178–82. [12] Karimi A, Shojaei OR, Kruml T, Martin JL. Characterization of TiN thin films using the bulge test and nanoindentation technique. Thin Solid Films 1997;308/309:334–9. [13] Tadmor EB, Miller R, Phillips R, Ortiz M. Nanoindentation and incipient plasticity. J Mater Res 1999;14:2233–50. [14] Nakamura T, Shih CF, Freund LB. Analysis of a dynamically loaded three-point bend ductile fracture specimen. Eng Fract Mech 1986;25:333–9. [15] Belytshko T, Liu WK, Moran B. Nonlinear finite elements for continua and structures. New York: Wiley; 2000. [16] Wriggers P, Vu Wan T, Stein E. Finite element formulation for large deformation impact-contact problems with friction. Comput Struct 1990;37:319–31. [17] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985. [18] Anderson TL. Fracture mechanics: fundamentals and applications. 2nd ed. London: CRC Press; 1995.
A numerical fracture analysis of indentation into thin hard films on soft substrates K. Sriram, R. Narasimhan *, S.K. Biswas Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India Received 4 December 2001; received in revised form 10 April 2002; accepted 30 April 2002
Abstract In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the mechanics of fracture of the film due to formation of cylindrical or circumferential cracks extending inwards from the film surface. Also, the role of plastic yielding in the substrate on the above mechanics is studied. To this end, the plastic zone development in the substrate and its influence on the load versus indentation depth characteristics and the stress distribution in the film are first examined. Next, the energy release rate J associated with cylindrical cracks is computed. The variation of J with indentation depth and crack length is investigated. The results show that for cracks located near the indenter axis and at small indentation depth, J decreases over a range of crack lengths, which implies stability of crack growth. This regime vanishes as the location of the crack from the axis increases, particularly for a substrate with low yield strength. Finally, a method for combining experimental load versus indentation depth data with simulation results in order to obtain the fracture energy of the film is proposed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thin hard films; Indentation; Finite elements; Fracture; Energy release rate
1. Introduction It is realised today that one of the most effective ways to protect an engineering component operating in erosive, corrosive or high temperature environment under heavy contact traction is to coat it thinly with a hard material like diamond or a ceramic. Hard coatings on soft substrates may be single or multi-layered epitaxial with thickness ranging from a few nm (as in the case of diamond-like carbon coatings used in magnetic recording) to 100 lm. In such situations, the mechanical integrity of the film should be preserved. The stresses resulting from contact tractions on thin films are generally very high. The hardness of the film enables it to resist plastic yielding that is easily induced in the underlying substrate even when the tractions are small. However, under these conditions, different failure modes may operate in the film. The
*
Corresponding author. Tel.: +91-80-394-2959; fax: +91-80-360-0648. E-mail address: [email protected] (R. Narasimhan).
0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0013-7944(02)00112-1
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first type of failure is associated with spalling of the film. With increasing traction, cracks which may be radial [1,2] or circumferential [3] extend deeper into the film and reach the film–substrate interface. If substantial yielding occurs in the substrate, the cracks may deflect and propagate along the interface [1]. There is a third possibility wherein the crack may extend into the substrate. The properties which need to be determined in order to design a coated system are the modulus and strength of the film and fracture toughness of both the film as well as the interface. The effectiveness of conventional techniques (such as micro-hardness test) to determine the properties of the film are severely limited due to the sheer smallness of the film thickness. Also, introduction of a submicrometric pre-crack for determination of the fracture toughness of the film is restricted due to limitations in fabrication. The peel test, scratch test and bending test may be conducted on a comparative basis, to yield a measure of adhesion of the film to the substrate [4]. However, its relationship to more fundamental properties such as strength and fracture toughness of the interface is as yet not clear. In this context, it must be mentioned that Wei and Hutchinson [5] have recently examined the mechanics of the peel test, while Volinsky et al. [6] have reviewed thin film adhesion and proposed relationships between film constitutive properties and interfacial toughness. In view of the above problems associated with the dimensional smallness of thin films, depth sensing nanoindentation has become a popular technique in recent times to provide quantification of the mechanical properties of the film and the interface. Thus, determination of the elastic modulus and hardness of a coated sample from nanoindentation load–displacement characteristics is now well established [7], given the area function associated with the indenter tip. However, methods for obtaining the fracture toughness of the film or interface from these test results are not well developed. Since in most structural applications, the film is brittle, this persists as an important obstacle to thin film design. One of the earliest approaches to resolving this problem was due to Marshall and Lawn [8] who gave a semi-empirical formulation for estimating the fracture toughness from data obtained during indentation by a Vickers indenter. However, this analysis is approximate and also does not take cognizance of the fact that crack growth may produce discontinuities or steps in the load–displacement characteristics. Such steps in nanoindentation test records have been observed experimentally [3,9]. A schematic of an experimental load versus indentation depth curve containing a step will be presented later in Section 4.5. The occurrence of these steps in load–displacement curves has been shown to be associated with the formation of circumferential cracks around the indented zone [3,10,11]. Further, these cracks grow along cylindrical surfaces through the thickness of the film [3,10]. Such type of cracking may indeed be a natural occurrence in films having a columnar grain structure with the axes of the elongated grains being perpendicular to the film surface [3,12]. However, the interpretation of experimental data to obtain the fracture toughness of the film has been based on approximate relationships [2] and a posteriori measurement of crack lengths from the coated sample which is a difficult task. It must be noted that emission of dislocations underneath the indenter may be responsible for the discontinuities in the load–displacement characteristics at indentation depths of the order of a few angstroms as discussed, for example, by Tadmor et al. [13]. This gives rise to strain localization and hardness gradients near the film surface. The continuum (fracture mechanics based) viewpoint adopted in this paper would be applicable only to situations where the film thickness and indentation depths are at least an order of magnitude more than the above size scales. The first step towards providing a sound fracture mechanics methodology for analysing indentation test results was taken by Weppelmann and Swain [3]. They conducted a detailed study of the stress fields that develop during spherical indentation of thin hard films deposited on soft substrates. The focus of their work was to investigate the effects of indenter radius to film thickness ratio and plastic yielding in the substrate on the above fields. The weight function method was employed to compute the stress intensity factors associated with cylindrical cracks in the film. However, a systematic investigation of the mechanics of fracture in the film during indentation was not performed in their work. Further, the use of the weight function
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method entails the assumptions of plane strain (whereas, in reality, the domain is axisymmetric), and small geometry changes in the indented zone. Hence, the primary objective of this paper is to investigate the mechanics of fracture in thin hard films on more ductile substrates due to occurrence of cylindrical cracks during spherical indentation. To this end, the stress fields in the film, as well as the energy release rate associated with cylindrical cracks at different radial locations and indentation depths are studied using finite element analyses. An important aim of this work is to assess the role of plastic yielding in the substrate on the above results. Hence, the J2 flow theory of plasticity is employed to describe the constitutive response of the substrate and different yield strengths are considered. The film is assumed to be isotropic and linear elastic and to be free of residual stresses. The energy release rate is computed from the finite element results continuously throughout the indentation analysis of films with pre-existing cylindrical cracks using the numerically accurate domain integral formulation [14]. Finally, a method is proposed by which finite element simulations can be employed to interpret indentation test results and yield information related to the fracture behaviour of the film.
2. Numerical procedure In this work, axisymmetric finite element analyses of spherical indentation of the film–substrate system is carried out. In the analyses reported in Section 3, the film is assumed as uncracked, whereas, cylindrical cracks extending inwards from the film surface around the zone of indentation are modelled in Section 4. An updated Lagrangian finite element formulation [15] is employed in all these analyses. A schematic of the axisymmetric geometry considered in the simulations undertaken in Section 3 is shown in Fig. 1. The ratio of the film thickness to indenter radius, t=Ri , is chosen as 1.5. The focus of the present work is on brittle cracking of the film and not on debonding of the film–substrate interface. Hence, the film is assumed to be perfectly bonded to the interface. The ratios of the radial dimension of the
Fig. 1. Schematic showing the film–substrate system along with the spherical indenter in the r–z plane.
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film–substrate system and the substrate thickness to the indenter radius, Rs =Ri and ts =Ri , are taken as 300 and 150, respectively, so that boundary effects do not influence the stress distribution near the zone of indentation. The contact between the (rigid) indenter and the film is assumed as frictionless and is modelled using a slideline approach along with a penalty formulation [16]. The film is assumed to be linear elastic, whereas a finite deformation, J2 flow theory of plasticity model [15] is employed for the substrate. The material properties are chosen to be representative of a TiN film on a steel substrate (as in the experiments reported in [3]). Thus, the ratio of the Young’s modulus of the substrate to that of the film, Es =Ef , is taken as 0.4. The Poisson’s ratios of the film and substrate, mf and ms , are assumed as 0.2 and 0.3, respectively. Further, the ratio of the tangent modulus to Young’s modulus, Ets =Es , of the substrate is chosen as 0.04. In order to study the effect of plastic yielding in the substrate on the indentation mechanics, three values of initial yield strain of the substrate, rys =Es ¼ 0:001, 0.003 and 0.009 are considered. The finite element meshes used in the analyses are comprised of four-noded isoparametric quadrilateral elements and are well refined near the zone of indentation. It must be noted that since a continuum analysis with homogeneous material properties is performed here, the results obtained would depend only on the ratios of different length dimensions and not on their absolute values. Thus, for example, from analytical solutions for contact between a sphere and an elastic half-space [17], as well as from dimensional considerations, it may be expected that the load P during indentation of a film bonded to a soft substrate (as in Fig. 1) would have the following functional form, h t Es rys P ¼ Ef R2i f ; ; ; ; other material constants : ð1Þ t Ri Ef Es Here, h=t is the ratio of the indentation depth to film thickness and is a measure of the compressive axial strain in the film at the indenter axis. It should be mentioned that additional length scale considerations may be important at submicron thicknesses where localized strain gradients in the substrate may become dominant.
3. Stress analysis of uncracked film–substrate system It is important to first understand the development of plastic yielding in the substrate during indentation and its influence on the load versus indentation depth characteristics and stress variations in the film. This would enable interpretation of the trends in variation of energy release rate with indentation depth and crack length in the fracture analysis to be performed later. Hence, a detailed analysis of indentation of an uncracked film–substrate system is undertaken in this section. 3.1. Load versus indentation depth The variation of normalized load P =ðEf R2i Þ with indentation depth h=t up to a maximum indentation depth of hm =t ¼ 0:05 and 0.2 are presented in Fig. 2(a) and (b), respectively. In these figures, the solid lines correspond to an initial yield strain of the substrate of rys =Es ¼ 0:009 and the dashed lines to rys =Es ¼ 0:001. Both the loading and unloading segments (starting from maximum indentation depth) are shown in these figures. It can be observed from Fig. 2(a) that for rys =Es ¼ 0:009, the loading and unloading curves coincide with each other, indicating a reversible (elastic) response of the film–substrate system. However, for rys =Es ¼ 0:001, the unloading curve differs from the loading curve even at hm =t ¼ 0:05 (see Fig. 2(a)), which is attributed to the development of plastic yielding in the substrate. As the normalized indentation depth is increased, it may be noticed from Fig. 2(b) that the unloading segments for both cases deviate substantially from the loading curves.
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Fig. 2. Normalized load versus indentation depth for rys =Es ¼ 0:001 and 0.009 corresponding to maximum indentation depth of (a) hm =t ¼ 0:05 and (b) hm =t ¼ 0:2. The unloading curves from maximum indentation depth are also shown.
It can be observed from Fig. 2(a) and (b) that the peak load at a given hm =t increases with the yield strength of the substrate. Also, the P versus h curve becomes almost linear at large h=t (see Fig. 2(b)). In s order to characterize this effect, variations of the type P =ðEf R2i Þ ¼ Aðh=tÞ , where A is a constant, were fitted to the loading curves. The values of the index s corresponding to different ranges of indentation depth are summarized in Table 1 for the three values of rys =Es . It must first be noted that the index s for spherical Table 1 Values of exponent s obtained by fitting curves of the type P =ðEf R2i Þ ¼ Aðh=tÞs to the numerical results corresponding to different rys =Es and range of h=t Range of h=t
rys =Es 0.001
0.003
0.009
0–0.05 0.05–0.1 0.1–0.2
1.4 1.2 1.0
1.46 1.29 1.13
1.48 1.4 1.26
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indentation of an elastic film bonded to a rigid substrate should be 1.5, whereas that pertaining to indentation of an elastic–perfectly plastic bulk solid under large scale yielding is 1 [17]. Now, from Table 1 it may be seen that at low h=t, s is close to 1.5 for all rys =Es which implies linear elastic Hertzian behaviour of the film–substrate system. As h=t increases, the value of s decreases to 1, which is an outcome of plastic yielding in the substrate. This is inferred from the fact that the index s approaches 1 more readily (i.e., at smaller indentation depths) when the substrate yield strength is less and there is considerable substrate plasticity (see Section 3.2).
3.2. Plastic strain contours Rt 1=2 The contours of equivalent plastic strain, p ¼ 0 ð2=3Dpij Dpij Þ dt, where Dpij is the plastic part of the rate of deformation [15], in the substrate at hm =t ¼ 0:2 are shown in normalized coordinates in Fig. 3. This figure pertains to the case rys =Es ¼ 0:003. It can be seen from this figure that the plastic strain contours extend more in the z direction. By contrast, at small h=t, it was found that they spread more in the radial direction. With further indentation, they become circular in shape and finally extend more in the z direction at large h=t (as in Fig. 3). Also, the level of plastic strain at a given distance from the point of intersection of the axis and the interface (i.e., r ¼ 0, z=t ¼ 1) increases with indentation depth. As discussed above, the elastic–plastic boundary in the substrate is not spherically symmetric as assumed in Johnson’s cavity model [17] for indentation of an elastic–perfectly plastic half-space. Hence, the normalized maximum extent of the substrate plastic zone from the point of intersection of the axis and the interface, rp =t, is plotted as a function of normalized indentation depth in Fig. 4. Results pertaining to three values of rys =Es are shown in this figure. It may be seen from this figure that the substrate plastic zone grows strongly with indentation depth, particularly for low rys =Es . Further, at any given indentation depth, the enhancement in rp =t with reduction in rys =Es is quite dramatic. This is not too surprising since the spherical cavity model of Johnson [17] also predicts a strong increase in the plastic zone size in an elastic–perfectly plastic bulk solid with decrease in yield strength.
Fig. 3. Contours of equivalent plastic strain in the substrate for the case rys =Es ¼ 0:003 at h=t ¼ 0:2.
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Fig. 4. Normalized substrate plastic zone size versus indentation depth for rys =Es ¼ 0:001, 0.003 and 0.009.
3.3. Stress distribution in the film The variation of normalized radial stress rrr =Ef along the film surface (i.e., z ¼ 0 in the undeformed configuration) corresponding to h=t ¼ 0:2 is shown in Fig. 5. Results pertaining to rys =Es ¼ 0:001 and 0.009 are presented in this figure. Here, the radial distance r is normalized by the contact radius a. The radial stress distributions displayed in Fig. 5 are typically Hertzian [17] and show a tensile peak just outside the contact zone. The magnitude of the tensile peak decreases with reduction in rys =Es (particularly at large h=t as in Fig. 5). The tensile peak implies that a cylindrical crack located just outside the contact zone and extending inwards (i.e., in the negative z direction) from the film surface is likely to be triggered (see also [3]). Hence, in the fracture analysis conducted later, the energy release rate associated with such cylindrical cracks at different indentation depths is studied.
Fig. 5. Normalized radial stress distribution along the film surface for rys =Es ¼ 0:001 and 0.009 at h=t ¼ 0:2.
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Fig. 6. Variation of normalized stresses through the film thickness for two values of rys =Es and r=a, where a is the contact radius: (a) rrr =Ef versus jzj=t at h=t ¼ 0:05, (b) srz =Ef versus jzj=t at h=t ¼ 0:05, (c) rrr =Ef versus jzj=t at h=t ¼ 0:2 and (d) srz =Ef versus jzj=t at h=t ¼ 0:2.
The variations of rrr =Ef and srz =Ef at two locations outside the contact zone are shown in Fig. 6(a) and (b) as a function of subsurface depth jzj normalized by t, corresponding to h=t ¼ 0:05. Similar stress distributions at h=t ¼ 0:2 are presented in Fig. 6(c) and (d). It can be seen from Fig. 6(a) and (c) that rrr =Ef decreases rapidly from the tensile value at the film surface and becomes compressive at a small distance jzj=t below the surface. It attains a peak compressive value between jzj=t ¼ 0:05 and 0.2 depending upon the radial location and indentation depth. The radial stress is small in magnitude for jzj=t > 0:4 at low indentation depth (see Fig. 6(a)). On the other hand, it may be noticed from Fig. 6(c) that at large indentation depth, rrr =Ef becomes tensile for jzj=t > 0:4. In fact, a strongly tensile radial stress prevails at the film–substrate interface. The tensile radial stress at the interface is higher for lower rys =Es , which along with the observation that it is pronounced at large h=t suggests that it is caused by plastic yielding in the substrate. It may be observed from Fig. 6(b) and (d) that for r=a > 1, srz =Ef increases as jzj=t increases, reaches a peak value between 0.05 to 0.3 depending upon r=a and h=t, and then decreases. At small distances jzj=t below the film surface, srz =Ef reduces strongly with increase in radial distance r=a, whereas at large jzj=t (greater than 0.4), it remains fairly constant as r=a changes from 1.1 to 1.5. Also, srz =Ef tends to zero as jzj=t approaches 1, irrespective of rys =Es and h=t, at least up to h=t ¼ 0:2. On comparing Fig. 6(b) and (d), it can be seen that as h=t increases, the magnitude of the peak srz =Ef increases for both values of r=a, and the location of the peak shifts to higher jzj=t. Further, Fig. 6(d) shows that at h=t ¼ 0:2, the magnitude of srz =Ef for a given r=a decreases substantially with reduction in rys =Es for the entire range of jzj=t from 0 to 1. Thus, while reduction in rys =Es elevates the tensile radial stress at the interface, it strongly reduces the shear stress at any distance jzj=t below the film surface. These trends affect the energy release rate pertaining to cylindrical cracks in the film as will be seen in Section 4.
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4. Fracture analysis The results of the stress analysis conducted in the previous section show that a peak tensile radial stress occurs on the film surface just outside the edge of the contact zone (Fig. 5). Further, as noted in the introduction, experimental results (see, for example, [3]) suggest that circumferential cracks will propagate along vertical surfaces during indentation of the film with spherically tipped indenters. Hence, the mechanics of fracture of the film due to the formation of such cracks is examined in this section.
4.1. Modelling aspects A schematic of the problem addressed in this section is illustrated in Fig. 7. The ratios of the film thickness and other geometrical dimensions to the indenter radius are taken as mentioned in Section 2. Further, as shown in Fig. 7, a pre-existing cylindrical crack of length c located at a radius R is modelled. The values of R=Ri that are chosen in this study are 0.27, 0.36 and 0.46. The maximum normalized indentation depths hm =t up to which the analyses are conducted for these three cases are 0.05, 0.1 and 0.2, respectively. It was found from the results obtained in Section 3 that the normalized contact radii a=Ri for an uncracked film at these indentation depths are slightly less than the three R=Ri values mentioned above. Also, it must be noted that the radius of the contact zone adjacent to the indenter axis at a given indentation depth will decrease due to the presence of a circumferential crack owing to the increase in the compliance of the film. In each of the above cases, fracture analysis is conducted for a range of crack length to film
Fig. 7. Schematic showing in the r–z plane a film with a cylindrical crack of length c, located at radius R, which is bonded to a soft substrate and subjected to spherical nanoindentation.
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thickness ratios, c=t, varying from 0.1 to 0.9. The constitutive response and properties of the film and substrate are chosen as indicated in Section 2. The compressive radial stress prevailing below the film surface (see Fig. 6) leads to crack closure for shallow cracks (irrespective of indentation depth), as well as for deep cracks at small indentation depth. In these cases, the contact between the crack faces is assumed to be frictionless and the associated constraints are imposed by the penalty approach [16]. Thus, shallow cylindrical cracks in the film are essentially subjected to mode II loading. On the other hand, there is opening of the crack faces due to tensile radial stress for deep cracks at large indentation depth and, hence, a mixed-mode state prevails near the tip under these conditions. 4.2. Variation of energy release rate with indentation depth The energy release rate J is computed using the domain integral method [14] from several rectangular domains of dimensions varying from 0:01 t 0:01 t to 0:125 t 0:125 t about the crack tip. It is found that J computed from various domains differ by less than 3% for all cases. In order to check the accuracy of these J values, the energy release rate is also obtained for a few cases using the load versus indentation depth curves pertaining to incrementally different crack lengths. In this context, it should be noted that for a cylindrical crack located at radius R from the axis of symmetry in a nonlinear elastic solid, the energy release rate can be defined in terms of the load P versus load point displacement D as follows [18]: ! Z D ~ ; cÞ 1 oP ðD ~: J ¼ dD ð2Þ 2pR 0 oc On applying this relation to the indentation problem, and interpreting D as indentation depth h, it may be argued that the hatched area dA shown in Fig. 8 between P versus h curves corresponding to cracks of
Fig. 8. Schematic of load versus indentation depth curves corresponding to cracks of length c and c þ dc. The shaded area gives J as indicated in the inset.
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Fig. 9. Comparison between normalized J versus h=t variations obtained from P–h curves and the domain integral method for a crack of length c=t ¼ 0:225 located at R=Ri ¼ 0:27, corresponding to rys =Es ¼ 0:003.
length c and c þ dc is given by dA ¼ 2pRJ dc. It must be emphasized that in defining J via Eq. (2), it is assumed that the material behaves as a nonlinear elastic solid. Thus, its application to elastic–plastic materials should be limited to situations where there is monotonic proportional loading, so that the constitutive response can be accurately represented by the deformation theory of plasticity (which is like a nonlinear elasticity model) [18]. In Fig. 9, the normalized J computed from the P versus h variations corresponding to cracks of slightly different lengths (by about 3%) is compared with that obtained from the domain integral method for a typical case. This figure pertains to R=Ri ¼ 0:27 and c=t ¼ 0:225. It may be seen from this figure that the energy release rates computed by the two methods agree closely with each other. Similar agreement was observed for cracks located at larger values of R=Ri and at higher indentation depths. This lends confidence to the energy release rate values obtained from the finite element simulations in this work. In Section 4.5, a method for obtaining the fracture energy Jf of brittle films is suggested by applying Eq. (2) to experimental P versus h records containing steps (which correspond to growth of cylindrical cracks). The variations of normalized energy release rate with indentation depth for cracks located at R=Ri ¼ 0:36 and corresponding to different normalized lengths c=t are shown in Fig. 10. This figure pertains to rys =Es ¼ 0:003. It was found from an analysis of the curves presented in Fig. 10 that J increases approximately as h2:5 for small h=t (up to 0.05), whereas it increases as h2 at large h=t. This change in trend is an outcome of the influence of substrate plasticity on the variation of load with indentation depth which was noted earlier. Thus, the overall fracture response of the film–substrate system is elastic at small h=t, whereas, it is governed by plastic yielding in the substrate at large h=t. Further, it was found that for shallow cracks (say, c=t < 0:4), J at a given h decreases dramatically as the radial location R=Ri of the crack surface increases. By contrast, J is quite independent of R=Ri , at least up to R=Ri ¼ 0:5, for deep cracks at a given h=t. The above trends can be rationalized from Fig. 6(b) and (d) which show a sharp decrease in srz with r=a at small jzj=t, whereas srz is fairly independent of r=a at large jzj=t. It follows that among a series of concentric cracks, the one which is closest to the symmetry axis will be triggered first if the crack length is small. On the other hand, all cracks located over a range of radial distances are likely to be triggered at around the same indentation depth when the crack length is large.
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Fig. 10. Normalized J versus h=t curves corresponding to different crack lengths for cracks located at R=Ri ¼ 0:36. The substrate yield strain is rys =Es ¼ 0:003.
4.3. Variation of energy release rate with crack length It can be observed from Fig. 10 that the energy release rate does not increase monotonically with crack length. In order to understand this effect clearly, the variation of normalized J with crack length c=t is shown in Fig. 11(a)–(c) for indentation depths of h=t ¼ 0:05, 0.1 and 0.2, respectively. These figures pertain to R=Ri ¼ 0:27, 0.36 and 0.46, respectively. In Fig. 11(a) and (b), only results corresponding to rys =Es ¼ 0:003 are presented, whereas in Fig. 11(c) those pertaining to three values of rys =Es are shown. On examining Fig. 11(a), it may be seen that for cracks located close to the symmetry axis and at small indentation depth (h=t ¼ 0:05), J =ðEf Ri Þ increases initially with crack length and reaches a peak at around c=t ¼ 0:2. It then decreases between c=t ¼ 0:2 and 0.75, and thereafter shows a marginal increase. The decreasing branch in the J versus c=t variation is caused by the reduction in the shear stress in the film for jzj=t > 0:2 as seen in Fig. 6(b). It can be observed from Fig. 11(b) that with increase in R=Ri and h=t, the range of crack lengths over which the reduction in J =ðEf Ri Þ occurs, as well as the magnitude of this drop relative to the peak value, are diminished. Finally, it can be seen from the curve pertaining to rys =Es ¼ 0:003 in Fig. 11(c) that at large R=Ri and h=t, the decreasing branch completely vanishes, although J remains almost constant over the range of c=t from 0.3 to 0.6. This is because, at large indentation depth, rrr becomes strongly tensile as the interface is approached (see Fig. 6(c)) which compensates for the reduction in srz with jzj=t. From Fig. 11(c), it can be observed that for any given crack length, the value of J is higher when rys =Es is larger. This counter-intuitive result may be rationalized as follows. First, it must be noted that yielding occurs in the substrate and not near the crack tip in this analysis. Further, it was found that the plastic work dissipated in the substrate is negligible at small indentation depth (h=t < 0:1) compared to the elastic energy of the film, whereas it is reasonably independent of rys =Es at large h=t. On the other hand, the elastic energy of the film increases with rys =Es which becomes pronounced at large h=t. Hence, the influence of rys =Es on J as seen in Fig. 11(c) is an outcome of its effect on the stress distribution in the film. It must be recalled from Fig. 6(d) that at h=t ¼ 0:2, the magnitude of srz is significantly higher for rys =Es ¼ 0:009, irrespective of jzj=t, which results in higher J for this case. Finally, it is interesting to note from Fig. 11(c) that for the case of rys =Es ¼ 0:009, J versus c=t shows a marginally decreasing branch between c=t ¼ 0:3 and 0.6 even at h=t ¼ 0:2.
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Fig. 11. Normalized J versus c=t for cracks (a) located at R=Ri ¼ 0:27 corresponding to h=t ¼ 0:05, (b) located at R=Ri ¼ 0:36 corresponding to h=t ¼ 0:1 and (c) located at R=Ri ¼ 0:46 corresponding to h=t ¼ 0:2.
4.4. Stability of crack growth The decreasing branch of the J versus c=t curves at low h=t for cracks located at small radial distances implies that crack growth over the corresponding range of crack lengths will be stable when the indentation test is carried out under displacement controlled conditions. In other words, the indentation depth will have to increase gradually in order to sustain further crack growth. From Fig. 11(c), it may be seen that the stable branch in the J versus c=t curve persists even for large R=Ri and h=t, if rys =Es is large. However, for low to intermediate strength substrates, growth of cracks located far away from the axis is likely to be unstable at large h=t over the entire range of c=t from 0 to 1 even under displacement controlled conditions since ðoJ =ocÞh > 0 (see Fig. 11(c)). The crack growth behaviour discussed above can also be understood by plotting the normalized P versus h=t curves corresponding to different crack lengths along with lines of constant J =ðEf Ri Þ as shown in Fig. 12. This figure pertains to rys =Es ¼ 0:003 and R=Ri ¼ 0:27. The lines of constant J =ðEf Ri Þ are obtained by connecting combinations of P and h pertaining to various c=t for which J =ðEf Ri Þ is equal to a chosen value. Fig. 12 shows that between c=t ¼ 0:1 and 0.2, the constant J trajectories correspond to decrease in both P and h which results in unstable crack growth under both load and displacement controlled conditions. In the range of c=t between 0.2 to 0.75, crack growth at constant J occurs with increasing h which gives rise to stable crack growth when displacement is prescribed in the indentation test. Finally, for c=t > 0.75, the constant J lines again correspond to decrease in both P and h which leads to unstable crack growth.
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Fig. 12. Normalized P versus h=t curves corresponding to different crack lengths along with constant normalized J trajectories for the case R=Ri ¼ 0:27. The substrate yield strain is rys =Es ¼ 0:003.
4.5. Interpretation of experimental P versus h data As mentioned earlier, experimental data of load versus indentation depth for thin films sometimes display steps which are associated with the formation of cracks in the indented region. These steps could be in the form of an increase in h at constant P or a sudden decrease in the slope of the P versus h curve. In this subsection, a method for interpreting such P versus h data and to obtain meaningful estimates of the fracture energy Jf of the film is presented. In Fig. 13, a schematic of an experimental P versus h curve with a step AB, which corresponds to the growth of a cylindrical crack at a radius R (similar to that observed by Weppelmann and Swain [3]), is shown. In order to obtain Jf from this experimental curve, the extent of crack growth Dc must be first determined. To this end, a family of P versus h curves corresponding to different crack lengths c1 , c2 , etc., at the observed radial location R (similar to Fig. 12), generated by finite element analyses, are superimposed on the experimental data as shown in Fig. 13. In this diagram, point A (beginning of the step in the experimental data) lies between the numerically generated P–h curves pertaining to crack lengths c1 and c2 (and closer to the latter). Similarly, point B may be observed to be close to the curve corresponding to crack length c3 . Hence, the extent of crack growth Dc associated with the step AB may be deduced to be around (c3 c2 ). The fracture energy Jf of the film may now be estimated in two ways. In the first method, Jf may be calculated as Af =ð2pRDcÞ, where Af is the area of the curved wedge OAB in Fig. 13. This approach is based on the application of Eq. (2) and assumes that the substrate material response is like a nonlinear elastic solid (see discussion in Section 4.2). The above assumption will be violated if there is significant elastic unloading in the substrate during crack growth by Dc in the film. In this context, it must be mentioned that this method can also be employed by considering a smaller curved wedge involving points O, A and a point D lying between A and B which is closer to A. Alternatively, Jf may be deduced by conducting further analysis of the results and plotting constant J lines corresponding to levels J1 , J2 , etc., along with the P versus h curves (similar to Fig. 12). Such constant J lines are shown by dash-dot curves in Fig. 13. It may be seen from this figure that the step AB in the experimental data lies close to the constant J line pertaining to the level J2 . Hence, Jf may estimated to be
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Fig. 13. Schematic of experimental load versus displacement data for a circumferentially cracked film (similar to that obtained in Ref. [3]), superposed on a computationally generated nomogram consisting of P versus h curves for different crack lengths and constant J trajectories.
equal to J2 . In fact, with more detailed analysis using the P versus h curves corresponding to different crack lengths and the constant J lines, the variation of Jf with crack length (due to say, small scale yielding in the film near the crack tip) may be deduced. 5. Conclusions The main conclusions of this work are listed below. 1. The load P increases as h3=2 and the unloading segment almost coincides with the loading curve for low indentation depth. This implies linear elastic behaviour of the film–substrate system. As the indentation depth increases, the exponent s characterizing the P versus h variation decreases and tends to unity. Also, the residual depth at complete unloading increases progressively. These trends are attributed to plastic zone development in the substrate. 2. The radial stress is tensile at the film surface outside the contact zone. It becomes compressive at a small distance jzj=t below the film surface. However, at large subsurface depth (jzj=t > 0:4), rrr once again becomes tensile. The tensile radial stress near the film–substrate interface (jzj=t ! 1) increases with h=t and is also higher for lower rys =Es at a given h=t. 3. The shear stress srz increases with distance jzj=t below the film surface, reaches a peak value between 0.05 and 0.3 (depending on the radial location), and then decreases. At large indentation depth, srz at a given distance below the film surface decreases substantially with reduction in rys =Es . 4. There is closure of crack faces for shallow cracks (irrespective of indentation depth) and for deep cracks at small indentation depth. Thus, under these conditions, the cracks are essentially subjected to mode II loading. On the other hand, the crack faces open out for deep cracks at large indentation depth leading to a mixed-mode state.
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5. For cracks located at small R=Ri and at small indentation depth, stable crack growth under displacement controlled conditions is expected to occur between c=t ¼ 0:2 and 0.75, irrespective of rys =Es . This stable crack growth regime vanishes as both R=Ri and h=t increase, albeit at a slower rate for a substrate with larger yield strength. 6. In order to interpret experimental P versus h data which show steps owing to formation of cylindrical cracks, it is suggested that this data be superposed on a computationally generated nomogram consisting of P–h curves pertaining to different crack lengths and constant J lines. This would enable determination of the extent Dc of crack growth, as well as the fracture energy Jf of the film.
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