A hypothesis is made that delamination can be driven by pockets of energy concentration (PECs) in the form of pockets of tensile stress and shear stress on and around the interface between a thin film and a thick substrate, where PECs can be caused by thermal, electrochemical or other processes. Based on this hypothesis, three analytical mechanical models are developed to predict several aspects of thinfilm spallation failure including nucleation, stable and unstable growth, size of spallation and final kinking off. The predictions from the developed models are compared against experimental results and excellent agreement is observed

Bin Wang (30851)

Christopher Harvey (1256223)

Simon Wang (1250952)

English

Loughborough University Institutional Repository

21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 THE MECHANICS OF INTERFACE FRACTURE IN LAYERED COMPOSITE MATERIALS: (5) THIN FILM SPALLATION DRIVEN BY POCKETS OF ENERGY CONCENTRATION – MICROSCOPIC INTERFACE FRACTURE  Bin Wang1, Christopher M. Harvey1 and Simon S. Wang1, 2  1 Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom E-mails: b.wang2@lboro.ac.uk, c.m.harvey@lboro.ac.uk, s.wang@lboro.ac.uk  2 School of Mechanical and Equipment Engineering,  Hebei University of Engineering, Handan, China  Keywords: Delamination, Pockets of energy concentration, Thermal residual stress, Spallation  ABSTRACT A hypothesis is made that delamination can be driven by pockets of energy concentration (PECs) in the form of pockets of tensile stress and shear stress on and around the interface between a thin film and a thick substrate, where PECs can be caused by thermal, electrochemical or other processes. Based on this hypothesis, three analytical mechanical models are developed to predict several aspects of thin-film spallation failure including nucleation, stable and unstable growth, size of spallation and final kinking off. The predictions from the developed models are compared against experimental results and excellent agreement is observed.  1 INTRODUCTION  Thin solid films are found in many different applications fulfilling various roles such as confine-ment of electric charge in integrated electronic circuits, thermal insulation in thermal barrier coatings (TBCs), and protection against corrosion, friction and wear in surface coatings. Although thin films are not usually expected to have a primary load-carrying capability, they often experience residual stresses due to the fabrication process and/or working conditions. Residual stresses are a major cause of film cracks and debonding. Buckling-driven delamination is a typical example of film failure under in-plane compressive residual stress, which has been extensively studied in the last few decades. How-ever, the experimental studies [1, 2] reveal a completely new failure behaviour of thin layer materials. A hypothesis is proposed in the present work to explain this behaviour. According to this hypothesis, delamination can be driven by PECs in the form of pockets of tensile stress and shear stress, with the former being dominant on and around the interface between a thin film and a thick substrate, where PECs can be caused by a number of different processes, including thermal cooling, chemical reactions and etc. Based on this hypothesis, circular-edged spallations are considered. Three mechanical models are established using partition theories for mixed-mode fracture based on classical plate theory, first-order shear-deformable plate theory and full 2D elasticity. 2 THEORY [3, 4] Fig. 1 shows a circular separation bubble of radius BR  of thin layer material of thickness h with the subscript B representing the edge of the bubble. The thin layer material is under in-plane biaxial compressive stresses 0 . The bubble shape is assumed to be radially sinusoidal and axisymmetric and represented by    BRrArw cos12 (1)  Bin Wang, Christopher M. Harvey and Simon S. Wang  Figure 1: A circular separation bubble of radius BR . with w  representing the upward deflection and A  the amplitude or the maximum separation. A clamped edge condition at BRr   is assumed because the thickness ratio between the thin layer and substrate is very small. The bubble energy is defined and calculated as    020220020111221RrBRrBa RhhRUUU  (2) in which U  is the total energy in the bubble including the bending strain energy, in-plane strain ener-gy and the interface fracture energy, and 0U  is the residual in-plane strain energy. E/)1( 200    and cGh 00    where E  and   are the Young’s modulus and the Poison’s ratio respectively of the thin layer material, and cG  is the interface fracture toughness. It can be shown that aU  is always posi-tive and monotonically increases with respect to the relaxation strain Rr  or the bubble amplitude A  when    022 12hRB . In this work, aU  is called ‘bubble energy’. It comes from the PEC energy formed in the form of pockets of tensile stress and shear stress, with the former being dominant on and around the interface between a thin layer material and the thick substrate. The PECs can be caused by a number of different processes including thermal cooling, chemical reactions and etc. When the PEC energy is able to provide the bubble energy aU  for nucleation, nucleation of a separation bubble will occur. It is expected that the bubble energy aU  governs the growth behavior of a bubble. By using the mixed mode partition theories [5-11] and a linear failure criterion, the bubble energy at bubble growth is obtained.     13623 2022202hRhRGRU BBcBGRa   (3) The first term in Eq. (3) is the sum of the bending strain energy and surface energy while the rest is the relaxed in-plane strain energy, which is negligible if BR  is small. The first term is therefore regarded as the nucleation energy, that is,   cBNUa GRU 25.1   where BR  is very small. It is seen that one third of the nucleation energy is used to bend the separation outwards after nucleating the interface separa-tion using two thirds of its energy. When the PEC is able to provide the bubble energy  GRaU , it will drive the nucleation and growth of a separation bubble. Two scenarios can occur: One scenario is slow and stable growth which occurs when BR  is smaller than the critical buckling characteristic length. The other is unstable growth when BR  reaches the critical value of the buckling characteristic length. The stable bubble then becomes an unstable buckle. The initiation of unstable growth is found at 21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017   2/12022 1112UGBhR (4) with the subscript UG denoting the initiation of unstable growth and 936.0 , and  cGEh221 2000   (5)   is the ratio between the plain strain energy density and interface fracture toughness. It will be seen later   plays key role for the mechanical behaviour of the bubble. There is no unstable growth when 2 . Expansion of the expression in the square bracket in Eq. (4) for 2 , leads to   02212UGBhR  (6) By differentiating  GRaU  in Eq. (3) with respect to hRB , its maximum is found to occur at    2/102289119MUBhR (7) with the subscript MU denoting the maximum. When 89  there is no solution. Expansion of the expression in the square bracket in Eq. (7) for 8/9 , leads to  0228MUBhR  (8) Note that  MUBR  in Eq. (8) is approximately equal to the critical buckling radius of a circular bubble with a clamped edge condition. Substituting Eq. (8) into Eq. (3) gives the bubble energy when 89  as   023223234364943 ccBcBMUa GhGRGRU   (9) More importantly,  GRaU  becomes zero at   2/102223116SPBhR (10) When 23  there is no solution. Expansion of the expression in the square bracket in Eq. (10) for 2/3 , leads to  0224SPBhR  (11) At this moment if the kinetic energy due to fast unstable growth of the buckle is large enough to break the film at its edge, the buckle spalls. The subscript SP in the equations above denotes spallation. The  Bin Wang, Christopher M. Harvey and Simon S. Wang final kink-off angle   can be determined as     210332arcsin  ccfGG (12) where cfG  is the fracture toughness of the thin layer material. Now, it is worth noting that the above development applies to all the three fracture mode partition theories, i.e. the classical, shear deformable plate [5-9] and 2D elasticity [10, 11] partition theories. However, the interface fracture toughness cG  should be, respectively for the three partition theories,  Icc GG   (13)  Icc GG  34  (14) and  Icc GG 6227.03773.0   (15) where IcG  is critical mode I ERR and IcIIc GG  . In the next section, predictions from the theory are compared against experimental observations from Refs. 1 and 2. 3 EXPERIMENTAL ASSESSMENTS Tolpygo and Clarke [1, 2] presented an excellent experimental study on the room temperature cir-cular spallation of α-alumina films grown by oxidation on Fe-Cr-Al alloy. The theory in Section 2 is applied to predict spallation behaviour [1, 2]. The material properties of the oxide film are taken from Refs. 1 and 2 and are as follows: The Young’s modulus of the oxide film is GPa 040E  and the Poisson’s ratio is 25.0 . The mode I critical ERR of the interface is mN 6.8IcG  and the critical mode I ERR of the oxide film is mN 20cfG . The ratio 5 IcIIc GG  is used, which is consid-ered by the authors to be a representative value. First, Eq. (6) is used to predict the initiation of unsta-ble growth, and Eq. (11) is used to predict the size of spallation. The solid dots in Fig. 2 represent a series of measurements of the size of individual separations as a function of time at room temperature. The time of 0 min corresponds to the moment when the specimen was placed under the microscope and its temperature was close to ambient. Fig. 2a shows data from four different separation bubbles on a single specimen after isothermal oxidation for 25 h at 1200°C and cooling at 20°C min-1. The bub-bles were successively monitored using optical microscopy. All of the bubbles grew at a constant compressive stress of GPa 4.046.40  , which was measured in the adherent oxide far away from the separations. The whole process includes nucleation, stable growth, unstable growth, and final spallation. The nucleation of separation bubbles was not recorded due to the difficulty of making time-ly observations of nucleating bubbles using this monitoring technique. Stable growth, however, with a radius far smaller than the critical buckling value, was readily observed. At a certain critical radius, which is again far smaller than the critical buckling radius, unstable growth abruptly occurs and final spallation takes place. It is pertinent that all four separations start unstable growth at approximately the same radius, and then all eventually spall off also at approximately the same radius. Two specimens with thicker oxide layers were produced with 50 h and 100 h of oxidation and are shown in Figs. 2b and c respectively. The growth behaviors of two separation bubbles are shown in Fig. 2b. Again, the two separations start unstable growth at approximately the same radius, and then both eventually spall off also at approximately the same radius. Fig. 2c shows the growth behavior of one separation bubble. 21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 The horizontal lines in Figs. 2a–c represent the predictions from Eq. (6) for the initiation of unstable growth and Eq. (11) for final spallation. It is very impressive to see that the predictions from Eqs. (6) and (11) have  Figure 2: Separation bubble radius versus time at room temperature for three different samples [2]. Excellent agreement with the test results. It is worth noting that Eqs. (6) and (11) are common to all three mechanical models (based on the classical plate, first-order shear-deformable plate and 2D elas-ticity partition theories) as long as the value of the parameter   meets the requirements of Eqs. (6) and (11). Next, comparisons are made between measurements of the kink-off angle and predictions from Eq. (12). The critical mode I ERR of the oxide film is mN 20cfG . For Fig. 2a, the value of   μm 10.9tan h  was measured approximately from Fig. 3 in Ref. [2], which is a similar case. The classical, shear deformable plate and 2D elasticity partition theories give  μm 5.7,μm 0.15,μm 8.3tan h , respectively. No test value for Fig. 2b was found in Tolpygo and Clarke’s studies [1, 2]. For Fig. 2c,   μm 46.13tan h  was measured approximately from Fig. 5 in Ref. (2), which is a similar case. The classical, shear deformable plate and 2D elasticity partition theo-ries give   μm 0.12,μm 9.23,μm 1.6tan h , respectively. The averages of the measurements of the  Bin Wang, Christopher M. Harvey and Simon S. Wang four diameters at 0°, 90° and ±45° were used to obtain the test values. It is seen that the 2D elasticity model gives good predictions but the other two models do not. It can be concluded at this point that the 2D elasticity model predicts the whole delamination pro-cess very well, including the initiation of unstable growth, size of spallation and kink-off angle. The other two models, however, only give good predictions of the initiation of unstable growth and the size of spallation. 4 CONCLUSIONS PECs can be formed by pockets of tensile stress and shear stress on and around the interface be-tween a thin film and a thick substrate, which can be caused by a number of different processes, in-cluding thermal effects, chemical effects and etc. PECs can cause interface spallation failure of thin films. Theory has been developed to predict several aspects of thin-film spallation failure by using par-tition theories for mixed-mode fracture based on classical plate theory, first-order shear-deformable plate theory [5-9] and full 2D elasticity [10, 11]. The three models all give accurate predictions of the initiation of unstable growth of separation bubbles and the size of spallation. The 2D elasticity model also gives accurate predictions of the final kink-off size but the classical plate and first-order shear-deformable plate models are unable to. The nucleation and stable growth of a separation bubble are solely driven by the bubble energy but unstable growth is driven by both bubble energy and buckling. Final kinking off is controlled by the toughness of the interface and the film and the maximum bubble energy. The 2D elasticity partition theory governs the microscopic brittle interface fracture instead of the classical partition theory which governs the macroscopic brittle interface fracture. The present me-chanical models reveal a new failure mechanism of thin layer materials under compressive residual stress and will be particularly useful to study the spallation failure of thermal barrier coating material systems. REFERENCES [1] V.K. Tolpygo and D.R. Clarke, Spalling failure of α-alumina films grown by oxidation. I. De-pendence on cooling rate and metal thickness, Material Science and Engineering, A278, 2000, pp. 142-150 (doi:10.1016/S0921-5093(99)00581-X). [2] V.K. Tolpygo and D.R. Clarke. Spalling failure of α-alumina films grown by oxidation. II. Decohesion nucleation and growth, Material Science and Engineering, A278, 2000, pp. 151-161 (doi:10.1016/S0921-5093(99)00582-1). [3] S. Wang, C.M. Harvey and B. Wang, Room temperature spallation of α-alumina films grown by oxidation, Engineering Fracture Mechanics, available online 14th March 2017 (doi: 10.1016/j.engfracmech.2017.03.002). [4] C.M. Harvey, B. Wang and S. Wang, Spallation of thin films driven by pockets of energy con-centration, Theoretical and Applied Fracture Mechanics, available on line 21st April 2017 (doi: 10.1016/j.tafmec.2017.04.011). [5] S. Wang S and C.M. Harvey, A theory of one-dimensional fracture. Composite Structures, 94, 2012, pp.758–767 (doi: 10.1016/j.compstruct.2011.09.011). Also a plenary lecture at the 16th International Conference on Composite Structures (ICCS16), 28–30 June 2011, Porto, Portugal. [6] S. Wang and C.M. Harvey, Mixed mode partition theories for one dimensional fracture, Engi-neering Fracture Mechanics, 79, 2012, pp. 329-352 (doi: 10.1016/j.engfracmech.2011.11.013). [7] C.M. Harvey and S. Wang, Mixed mode partition theories for one dimensional delamination in laminated composite beams, Engineering Fracture Mechanics, 96, 2012, pp. 737-759 (doi: 10.1016/j.engfracmech.2012.10.001). [8] C.M. Harvey and S. Wang, Experimental assessment of mixed-mode partition theories, Compo-site Structures, 94, 2012, pp. 2057-2067 (doi: 10.1016/j.compstruct.2012.02.007). 21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 [9] C.M. Harvey, M.R. Eplett and S. Wang, Experimental assessment of mixed-mode partition the-ories for generally laminated composite beams, Composite Structures, 124, 2015, pp. 10-18 (doi: 10.1016/j.compstruct.2014.12.064). [10] J.W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials, Advances in Applied Mechanics, 29, 1992, pp. 63-191 (doi:10.1016/S0065-2156(08)70164-9). [11] C.M. Harvey, J.D. Wood, S. Wang and A. Watson, A novel method for the partition of mixed-mode fractures in 2D elastic laminated unidirectional composite beams, Composite Structures 116, 2014, pp. 589-594 (doi: 10.1016/j.compstruct.2014.05.041). 

The mechanics of interface fracture in layered composite materials: (5) thin film spallation driven by pockets of energy concentration – microscopic interface fracture

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The mechanics of interface fracture in layered composite materials: (5) thin film spallation driven by pockets of energy concentration – microscopic interface fracture

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