Comments on the paper “Modification of composite hardness models to incorporate indentation size effects in thin films”, D. Beegan, S. Chowdhury and M.T. Laugier, Thin Solid Films 516 (2008), 3813-3817

Since the original work of Bückle concerning the substrate influence on the hardness measurement of thin film, more than 20 models were proposed to separate the contribution of the substrate. Subsequently to the development of these numerous models, a question arises: Which is the most relevant models among them? Indeed, the authors usually consider that their proposed model leads to the best prediction of the film hardness which is probably correct for a given experimental condition applied to a particular material. In addition, the authors also assume that the other models are not so relevant. But to have a sound discussion about the existing models, it is necessary to correctly apply them according to the author statement. In this paper, we better specified the application of the Jönsson and Hogmark model and that of Chicot and Lesage applied to the results obtained on copper films by Beegan et al. Contrarily to these authors, we show that the above-mentioned models lead to a good representation of the experimental data and a good predicted value of the film hardness.     

Analysis of data from various indentation techniques for thin films intrinsic hardness modelling

Due to the influence of the substrate, direct measurement of the hardness of thin films by standard micro-indentation tests is not always possible. In such situation, determination of the intrinsic film hardness requires the analysis of a set of experimental apparent hardness values obtained for different indentation loads. A number of mathematical equations based on various assumptions were proposed in literature for that purpose.Most of the models were established on the basis of standard Vickers indentation. Using these models to process the data obtained by Knoop indentation does not provide the same intrinsic hardness value, even after Knoop/Vickers standard conversion, than the one obtained from Vickers indentation. The same problem arises when processing the data coming from depth-sensing indentation. A method to obtain comparable hardness values is proposed in the present work by considering an “equivalent” Vickers hardness in the case of Knoop indentations and the corresponding Martens hardness for depth-sensing indentation. This method has been used to determine the intrinsic hardness of titanium nitride film.     

Analysis of ISE in dynamic hardness measurements of β-Sn single crystals using a depth-sensing indentation technique

The dynamic indentation tests on (001) plane β-Sn single crystals having different growth directions under different peak indentation test load (10, 20, 30, 40, and 50 mN) has been investigated. The experimental results reveal that the measured hardness values exhibit a peak-load dependence, i.e., indentation size effect (ISE). Such peak-load dependence is then analyzed using the Meyer's law, the Hays–Kendall's approach, the Elastic/Plastic Deformation model, the Proportional Specimen Resistance (PSR) model, and the Modified PSR (MPSR) model. As a result, Modified PSR model is found to be the most effective for dynamic hardness determination of β-Sn single crystals.     

A multilayer model for describing hardness variations of aged porous silicon low-dielectric-constant thin films

This paper reports on the micro-instrumented indentation of a porous silicon structure obtained by anodization of a highly p⁺-doped (100) silicon substrate aged over 1 week. The three-layer structure obtained consists of oxidized porous silicon (cap-layer), porous silicon (inner-layer) and silicon substrate. The hardness curve has the typical “U shape” of low-dielectric-constant films when the indentation depth rises: the early decrease in hardness, due to the soft inner layer, is followed by an increase, due to the hard substrate. A multilayer model is developed to account for hardness variation with respect to the applied load. This model considers the crumbling of the cap-layer and of the inner porous structure. As a result, it is shown that considering the minima in the U shape gives an over-estimated value when it comes to assessing the coating hardness. In our experiment, this minimum depends on both the hardness and the thickness of the oxidized cap layer, but not on the mechanical properties of the substrate, even for indentation depths slightly lower than the film's thickness.     

Extracting thin film hardness of extremely compliant films on stiff substrates

A previously reported method for extracting the thin film hardness from nanoindentation into a film on an elastically mismatched substrate was applied to four different cases of extreme mismatch in elastic properties: Parmax, Ultem, Polysulfone and Perfluorocyclobutyl polymer thin films on Si substrates. All of these cases represent extremely compliant films on a stiff substrate, where the ratio of film shear modulus to substrate shear modulus ranged from 0.008 to 0.036. Analyzing the nanoindentation data into these film/substrate systems poses a significant limitation when using the Oliver and Pharr method as the hardness increases rapidly with indentation depth. Therefore, a method involving the measured contact stiffnesses to more accurately determine the correct contact areas was used to extract the true hardness of the polymer thin films. The results indicate that our method is able to remove the substrate effects as well as the complications arising from pile-up and surface roughness to yield a wide plateau in hardness despite the extreme elastic mismatch conditions.     

A numerical fracture analysis of indentation into thin hard films on soft substrates

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.     

An analytical assessment of the influence of skin imperfections on the indentation collapse mechanism in composite sandwich beams

The influence of skin imperfections, in the form of delamination damage or thickness variations, on the indentation collapse mechanism in composite sandwich beams with compressive yielding cores is studied using the models of non-prismatic beam and beam-column resting on a nonlinear Winkler foundation. Upper and lower threshold solutions are derived for the indentation response and collapse load and the transition between the two limits is defined as a function of size, magnitude and position of the imperfections. In beams where global bending effects are not negligible, the collapse load is limited from above by the indentation collapse load of beams with rigid-plastic cores and the face wrinkling collapse load of beams with elastic cores; the transition between the two limits is controlled by material/structure properties and the magnitude of the imperfections. Characteristic lengths, which depend on material and geometrical properties, define the minimum size of the imperfections with the strongest effect on the solution and the minimum distance between load and imperfections with no effect on the solution.     

Interface stress induced hardness enhancement and superelasticity in polytetrafluoroethylene/metal multilayer thin films

Polytetrafluoroethylene (PTFE)/Al, PTFE/Cu, and PTFE/Ti multilayer thin films have been deposited in order to investigate effects of interface energy on mechanical properties. PTFE, which has a low surface energy of 19.2 mJ/m², was used to introduce a large interface energy into multilayer thin films. PTFE thin film was deposited by rf magnetron sputtering using a PTFE sheet target. Al, Cu, and Ti were deposited by dc magnetron sputtering. The multilayer thin films were fabricated sequentially without breaking vacuum. Substrate used was aluminosilicate glass. The modulation period was changed from 6.7 to 200 nm. The total thickness was about 200 nm for all samples. The internal stress of metal layers changed from tensile to compressive and increased with decreasing modulation period for all of PTFE/Al, PTFE/Cu, and PTFE/Ti. Both hardness enhancement and superelasticity were observed in the results of nanoindentation measurements. The energy dissipated during nanoindentation process (one load and unload cycle) decreased with decreasing modulation period. The minimum value of the ratio of dissipated/loaded energy was < 40%, which is smaller than the values obtained for monolithic PTFE or metal films (about 73% for PTFE and 87% for Al, 72% for Cu, and 71% for Ti, respectively). This meant that the PTFE/metal nano-multilayer thin films became more elastic with decreasing modulation period. The tendency of change in the mechanical properties strongly correlated to internal stress. Mechanisms involved in anomalous behaviors in film hardness and elasticity were discussed based on the relationship to interface energy, interface stress, and internal stress, induced by multilayering of the films. It is concluded that a large compressive stress introduced in the thin films increased the energy needed to deform elastically or plastically the thin film during indentation, resulting in the increase in hardness and elasticity. The nanoindentation analysis of the multilayer thin films emphasized that in PTFE/metal multilayer thin films mechanical properties of the films depend on interface stress induced by the accumulated interface energy, being independent of bulk materials properties composing thin films, resulting in increase in hardness and elasticity.     

A numerical analysis of flexure induced cylindrical cracks during indentation of thin hard films on soft substrates

In this paper, the fracture behavior of a thin hard film, perfectly bonded to a soft substrate, containing circumferential (cylindrical) cracks subjected to spherical indentation is studied using the finite element method. These cracks emanate upwards from the film-substrate interface and are driven by the flexure of the film over the soft substrate under indentation. The film is taken to be linear elastic while the substrate obeys an elastic-plastic constitutive model with linear strain hardening. Three values for the substrate yield strength are considered in the analysis. The variation of energy release rate and mode mixity are examined as functions of crack length and load, for cracks located near and away from the indentation axis. The results show that, when the crack length is small, predominantly mode I conditions prevail due to tensile radial stresses near the interface. As the crack length increases, the mode mixity gradually changes from mode I to II. For cracks located near the axis, the crack growth process is stable over a range of crack lengths up to about a third of the film thickness and thereafter becomes unstable. The role of the substrate yield strength on the above issues is investigated.     

Microstructure and hardness of SiC-TiC nanocomposite thin films prepared by radiofrequency magnetron sputtering

Silicon carbide-titanium carbide (SiC-TiC) nanocomposite thin films were prepared by radiofrequency magnetron sputtering using SiC-TiC composite targets fabricated by spark plasma sintering. The SiC thin films were amorphous at substrate temperatures below 573 K and crystallized in the cubic crystal system (3C) at substrate temperatures greater than 773 K. Cubic SiC-TiC nanocomposite thin films, which contain a mixture of 3C-SiC and B1-TiC phases, were obtained at a TiC content of greater than 20 mol%. The amorphous films possessed a dense cross-section and a smooth surface. The morphology of the SiC-TiC nanocomposite thin films changed from granular to columnar with increasing substrate temperature. The SiC-TiC nanocomposite thin films prepared at TiC content of 70-80 mol% and substrate temperature of 573 K showed the highest hardness of 35 GPa.     

Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies

Previous results of the author on the influence of size and boundary-conditions on the apparent properties of elastic heterogeneous materials are recalled and extended to the viscoelastic case. Materials with random microstructures, possibly with anisotropy, defects or damage, are considered. In place of the classical concept of effective properties, which holds for the representative volume only, the concept of apparent properties is recalled. It makes use of special kinds of boundary-conditions and appropriate stochastic averages for which the required Hill condition is still valid while the ergodic assumption – legitimating the use of the concept of equivalent homogeneous medium – is relaxed. Statistical apparent properties are defined on various partitions of a given initial specimen D₀ into a set of coarse specimens on the one hand, and of smaller specimens on the other hand. The boundary-condition and size-effects hierarchies derived in Huet (1990. Journal of Mechanical Physics and Solids 38, 813; 1997. Engineering Fracture Mechanics, (special issue) 58 (5–6) 459) are recalled. Examples of numerical verification by micromechanical simulations are shown using 3D finite element models of elastic granular composites. Some salient features of the viscoelastic behavior of non-metallic construction materials used in civil engineering are recalled. 3D simulations on viscoelastic micromechanical models are also shown, with results exhibiting trends similar to the elastic case and for which theoretical justification and generalisation is seeked in the paper. Use is made of a Statistical Continuum Thermodynamics approach through which the concept of equivalent homogeneous medium and effective properties is first discussed for materials with constituents having dissipative constitutive equations of any kind and from which natural definitions of dissipative potential and complementary energies are derived. Dissipative extensions of the so-called Hill condition are obtained. Then, new minimum theorems for viscoelasticity are derived from a slight modification of the pseudo-convolutive procedure used in Huet (1992. European Journal of Mechanics, A/Solids 5 (11) 653; In: Bazant, Z.P., Carol, I. (Eds.), Creep and Shrinkage of Concrete. Spon, London, pp. 189–200). From this, theoretical bounds and size-effect hierarchies on the statistical viscoelastic stiffness and compliance function tensors and their rates are obtained in the time domain. It is found that viscoelastic heterogeneous bodies smaller than the representative volume exhibit, for the two classes of uniform boundary-conditions, the same kind of size-effects as the elastic ones. Possible extensions to other kinds of time-dependent physical properties are quoted.     

Effect of surface step on nanoindentation of thin films by multiscale analysis

Nanoindentation simulations on flat and stepped surfaces are respectively investigated using the quasicontinuum method based on the embedded-atom method potential. Effect of surface step considering indenter size and step height is studied. Results show that the critical load for the first dislocation emission will be decreased with the increase of step height. However, the effect of surface step will be weakened if the indenter size continues to increase. Initial atomistic structures after dislocation nucleation and emission are discussed systematically. The initial dislocations are not quite identically nucleated under the stepped surface. Stress distribution analysis reveals that the shear stress in the slip planes close to the step is much larger than the shear stress in the slip planes far from the step for nanoindentation on the stepped surface. The multiscale simulation results are consistent with experimental results and analytic solutions. The conclusions about step effect considering indenter size and step height are helpful for understanding the microscopic mechanism of nanoindentation tests on thin films with surface step.     

Study of indentation size effect and microhardness of SrLaAlO<Subscript>4</Subscript> and SrLaGaO<Subscript>4</Subscript> single crystals

The load dependence of the Vickers microhardness of SrLaAlO₄ and SrLaGaO₄ single crystals, using a PMT-3 hardness tester, has been investigated and analysed from the standpoint of various theoretical models. On the (100) and (001) planes of these crystals, reverse indentation size effect was observed. Analysis of the experimental data revealed: (1) the indentation size effect is best described by Meyer's law and the proportional specimen resistance model of Li and Bradt, (2) indentation-induced cracking model for reverse indentation size effect and Meyer's law cannot be used to determine the hardness of the crystals, (3) as shown by the negative values of the load-dependent quantities in Hays-Kendall's approach and Li-Bradt model, the origin of indentation size effect is associated with the processes of relaxation of indentation stresses, and (4) the load-dependent and load-independent quantities of different models are interrelated and are intimately connected with the orientation and chemical composition of the crystals. It was also found that the plots of the ratio of indentation load to indentation diagonal against indentation diagonal for a sample exhibit two different slopes with a transition in the slopes occurring at an indentation diagonal, whose value depends on the indenter orientation, indented plane and chemical composition of the crystals. The physical significance of the appearance of these transitions and the nature of load-independent indentation microhardness are discussed. Electronic Publication     

Failure analysis of low velocity impact on thin composite laminates: Experimental and numerical approaches

The dynamic behavior of composite laminates is very complex because there are many concurrent phenomena during composite laminate failure under impact load. Fiber breakage, delaminations, matrix cracking, plastic deformations due to contact and large displacements are some effects which should be considered when a structure made from composite material is impacted by a foreign object. Thus, an investigation of the low velocity impact on laminated composite thin disks of epoxy resin reinforced by carbon fiber is presented. The influence of stacking sequence and energy impact was investigated using load–time histories, displacement–time histories and energy–time histories as well as images from NDE. Indentation tests results were compared to dynamic results, verifying the inertia effects when thin composite laminate was impacted by foreign object with low velocity. Finite element analysis (FEA) was developed, using Hill’s model and material models implemented by UMAT (User Material Subroutine) into software ABAQUS™, in order to simulate the failure mechanisms under indentation tests.     

纤维增强复合材料层合板缺口尺寸及形状效应数值模拟

通过考虑基体裂纹、纤维断裂、层内劈裂和层间脱层等破坏形式,建立三维有限元模型研究含中心圆孔和中心裂缝的准各向同性复合材料层合板([45/0/-45/90]_(2S))在拉伸载荷下的缺口尺寸效应及缺口形状效应。模拟结果显示:随着缺口尺寸的增大,层合板的破坏强度逐渐降低,然而,在本文研究范围内含中心裂缝的层合板破坏强度始终高于对应的含中心圆孔的层合板破坏强度。进一步分析有限元模拟结果表明,含中心裂缝的层合板亚临界损伤发生得更早,并且亚临界损伤范围更大,亚临界损伤会大大缓解缺口尖端的应力集中,从而使含中心裂缝层合板表现出更高的破坏强度。     

Cumulative size effects in electrical conductivity of thin semi-metal films

In the general framework of conduction models for monocrystalline and polycrystalline metal films, analytical expressions are derived for the electrical conductivity of semi-metal films. The existence of two cumulative size effects is thus predicted. These new equations agree with those previously derived in a special case of energy dependence of the relaxation time.     

Conductive composite films composed of polyaniline thin layers on microporous polyacrylonitrile surfaces

A facile approach for preparing flexible and conductive composite films by chemical polymerization of aniline monomers on the microporous polyacrylonitrile (PAN) surfaces was reported. A good adherence between polyaniline thin layer and PAN substrate was ascribed to the formation of a continuous conductive network in the film thickness along the capillary channel within PAN matrix. The electronic properties of the resulting composite films were tuned easily with the mass fraction of aniline monomer in the PAN solution or through chemical process. It is also found that the thermal stabilities of the composite films were significantly enhanced, while good mechanical properties were maintained.     

二元合金薄膜的表面偏析

本文评述了广泛应用于描述平衡态和动态偏析的修正Darken模型。该模型首次提出了偏析的驱动力是化学势梯度,并成功地应用于定量描述体块材料的平衡态和动态的表面偏析。在考虑到薄膜体系中的尺寸效应后,引入了一个约束条件,再将其应用于修正的Darken模型,实现了在所有的浓度和薄膜厚度范围内,对合金薄膜体系的平衡态和动态的表面偏析的模拟计算。并利用该模型模拟了N(i111)Cu二元合金薄膜系统的平衡态和动态的表面偏析。     

Effects of deposition parameters on hardness and lubrication properties of thin MoS2 films

The purpose of the present study is the development of a sputtered MoS₂‐coating suitable for the use in humid atmospheres. Influences of the process parameters argon pressure, temperature, distance between substrate and target and bias voltage on tribological properties and hardness are presented. By means of these parameter studies, different microstructures with widely varying wear behavior were achieved. Life cycle tests show that exclusively basal oriented coatings with dense, non columnar structures show the highest durability and a hardness of about 400 HV 0.01. However, dendritic needle like structures show lower endurance and lower hardness (<100 HV 0.01). Films with a comparatively high hardness (up to 800 HV 0.01) exhibit lowest sliding distances until failure due to an appearance of spalling in the ball‐on‐disc tests. Those coatings also show a dense but amorphous structure and sulfur deficiencies. Consolidation during sputtering, i. e. film hardness and porosity, can be controlled by certain process parameters and is probably affected by varying amounts of internal stress.