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On scratch testing of pressure-sensitive polymeric coatings.

Abstract Scratching of polymeric coatings on hard substrates is studied experimentally and numerically from a mechanical point of view. In particular, the behavior of local and global mechanical properties, as well as the delamination behavior, at scratching due to pressure-sensitive effects is of interest. The numerical investigation is performed using the finite element method (FEM) where pressure sensitivity is modeled using the classical Drucker-Prager plasticity model (Drucker, Prager, Q. Appl. Math., 10 157-165 (1952)) while the adhesion of the coating to the substrate is modeled as a cohesive zone where relevant model parameters are determined experimentally using the double cantilever beam test with uneven bending moments (DCB-UBM). Good correlation between experimental and numerical results were achieved and the most important finding concerns the fact that the dependence of pressure sensitivity proved to be different for different mechanical quantities.

Keywords Scratch testing, Film/substrate composites, Polymers, FE analysis, Pressure-sensitivity

Introduction

From a practical standpoint, it is of course of great importance to characterize thin films constitutively. Among the most widely used methods to perform the difficult task to investigate the material properties, indentation testing has been applied for a broad range of materials when determining the strength very near the surface. There are, however, very few theoretical methods available for the study of thin film indentation. Analytical methods can at best be used for studying elastic indentation of thin films but at elastoplastic indentation it is necessary to use more advanced methods. Accordingly, given the complexity of such an indentation analysis, numerical methods such as the finite element method (FEM) are necessary tools for determining relevant quantities in an accurate manner. The early studies by Bhattacharya and Nix, (1), (2) Laursen and Simo (3) and Giannakopoulos and co-workers, (4-7) demonstrated the usefulness of modeling a typical sharp indentation test using FEM and obtaining relevant values on, for example, the hardness in the case of bulk and thin film type samples. Furthermore, as regards sharp indentation, many authors reported on finite element studies performed on several thin film and substrate combinations, cf. e.g., references (8-11), and, perhaps more generally, in reference (12).

Based on the above discussion, it can then be concluded that, even though more study is required for a full understanding of the problem, substantial efforts have been directed toward the understanding of the mechanical behavior at thin film indentation. Today, scratch testing also is a well-established technique for hardness testing. Historically, one of the earliest efforts was Mohs' hardness scale from 1822, a scale based on the fact that a harder material will leave a visible scratch on a softer material if they are rubbed together. Throughout the coating industry, the scratch test is used for adhesion testing of coatings, and such a test usually consists of a loaded diamond tip that is drawn across a surface under continuously (or stepwise) increasing load. At some load, a well-defined failure occurs, and the critical load is found. This technique is used for the ranking of coating adhesion.

The fundamental knowledge about the mechanical behavior at scratching is, however, not nearly as developed as for indentation testing. It goes almost without saying that owing to the complexity of the boundary value problem, the analysis of this type of hardness testing also requires numerical methods, where again FEM is to be preferred, for a high accuracy analysis. In recent years, quite a few such analyses have been presented. Bucaille et al. (13) analyzed cone scratching of perfectly plastic materials and found, among other things, that a normalized strain measure, the well-known Johnson parameter (14) (to be explicitly shown below), can be used to correlate scratching experiments (it is well known from presiously, cf. e.g., reference (15), that this feature applies at normal indentation testing). This finding was later confirmed for strain-hardening elastoplasticity, by Wredenberg and Larsson. (16), (17)

Also when it comes to the analysis of thin film scratching, some investigations have been presented. These investigations have been devoted to severe scratching, i.e., fracture, delamination etc., cf. e.g., reference (18), as early as 1960, (19-23) as well as attempting to determine a more general understanding of the behavior of different global (and also local) scratch variables at these types of problems, cf. e.g., Larsson and Wredenberg (24) and Wredenberg and Larsson. (25)

When performing scratch experiments, a number of phenomena may complicate the evaluations of the experiments. Estimating the scratch depth can prove difficult due to difficulties in detecting initial contact, at least when using a conical stylus. At scratching of polymeric materials, time dependence may come into effect (26) both during the actual scratching and afterward when the residual deformation is to be measured. Also at scratching of polymers, the deformation is to a great extent elastic, making the estimation of the actual contact area based on the residual groove difficult. (26) From a practical point of view, a very important aspect of scratching of thin film/substrate systems concerns delamination along the film/substrate interface. This matter has been rather extensively investigated for hard coatings but not so for soft films (perhaps, the only exception being results by Yueguang et al. (27) pertinent to the "scraping" of a plastic lamella off a substrate). Possible reasons for this are the inherent difficulties associated with an analysis relating the delamination load to the elastic plastic deformation around the stylus. A number of the issues mentioned above were addressed in a parallel study by Wredenberg and Larsson (28) where delamination of soft polymeric coatings on hard (rigid) substrates was analyzed using FEM and cohesive zone modeling. In this article also, some preliminary results concerning pressure-sensitive behavior of polymeric materials at scratching were discussed. It is well known that many polymers exhibit pressure dependent plastic flow, cf. e.g., Spitzig and Richmond. (29), (30) Considering the very high plastic strain levels at scratching, this could be a feature that influences the results substantially, and it is the aim of this article to investigate this issue further.

Accordingly, the behavior of local and global properties, as well as the delamination mechanism, at scratching of thin polymeric homogeneous films, undergoing large deformations and high levels of plasticity, was analyzed numerically and experimentally with effects due to pressure-sensitive flow being of particular interest. In doing so, for obvious reasons when polymeric films are at issue the deformation of the substrate was neglected. Accordingly, the results of this study cannot be immediately extended to coating systems with hard and/or nonhomogeneous coatings. However, in case of hard, or more correctly harder, coatings some guidance is given in reference (25) as regards the validity of the assumption of a stiff substrate. For the experiments then, a previously experimentally characterized (17) Vinyl-Ester plate was machined to a thin film and adhered to a steel substrate using an epoxy resin. This was done to minimize the residual stresses (31) and material property gradients intrinsic to the polymerization of a film directly to the substrate. Additionally this enabled testing of the mechanical properties of the film material in bulk, which can be difficult if the film material exists only in a "film state". Furthermore the scratch procedure was simulated numerically using FEM where the adhesion of the film to the substrate was represented by a cohesive zone. The numerical simulation allowed for the variation of cohesive properties of the film substrate interface as well as for the mechanical properties of the film itself, resulting in a relatively extensive parametric study. The use of a cohesive zone model allows for the definition of a separation energy, [G.sub.c], necessary for the delamination of the film. For clarity and convenience, but not out of necessity, the analysis was restricted to cone scratching ([beta] = 22[degrees] in Fig. 1). In such a case with classical elastoplastic Drucker-Prager materials (32) accounting for pressure-sensitive flow, the indenter does not introduce any characteristic length in the problem,. In this context it should also be mentioned that [beta] = 22[degrees] was chosen as being a representative value for most sharp indenter geometries of practical interest.

Theoretical considerations and numerical analysis

The present analysis concerns scratching of thin film/substrate systems using a sharp conical stylus assumed to be rigid. It is assumed that quasi-static conditions prevail. In the equations below F and A represent contact load and contact area respectively, indices n and t represent the normal and tangential components of these quantities and h is the scratch depth shown in Fig. 1. It should be emphasized that the contact area A is, if not stated otherwise, the true projected contact area given by the numerical simulations.

[FIGURE 1 OMITTED]

Assuming that quasi-static and steady-state conditions prevail the monolithic scratch problem is self-similar with no characteristic length present. Consequently, the normal hardness

[H.sub.n] = [F.sub.n]/[A.sub.n], (1)

and the tangential hardness,

[H.sub.t] = [F.sub.t]/[A.sub.t], (2)

as well as the ratio h/[square root of]A are constant during the loading sequence of a scratch test on monolithic materials and stresses and strains are functions of the dimensionless variables x/[square root of]A (the Cartesian coordinate system is shown in Fig. 2) and material properties alone. At scratching (or normal indentation) of thin film/substrate systems, this is of course not the case at increasing load as the field variables then are also dependent on the ratio h/d, d being the film thickness as shown in Fig. 1. However, when the indentation depth is held constant during the test steady-state conditions prevail in the absence of cracking.

[FIGURE 2 OMITTED]

In this analysis, the material behavior of interest is pressure-sensitive flow which is here described using the Drucker-Prager (32) material model. The pressure sensitivity of such a material is characterized by (together with the standard assumptions in von Mises classical large deformation elastoplasticity) a parameter known as the "friction angle", [alpha], allowing the yield stress to vary with the mean stress [[sigma].sub.m]. The yield surface (see Fig. 3) may then be described as

[[sigma].sub.e] + tan([alpha]) [[sigma].sub.m] = [[sigma].sub.0]. (3)

[FIGURE 3 OMITTED]

In equation (3), [[sigma].sub.0] is the yield stress at zero mean stress (pure shear) and [[sigma].sub.e] is the Mises effective stress. Evidently, the angle [alpha] represents the slope of the yield surface in the [[sigma].sub.e] vs [[sigma].sub.m] stress space as shown in Fig. 3. It should be mentioned, though, that some of the numerical calculations were performed assuming classical von Mises plasticity (with isotropic hardening and accounting for large deformation) to apply. At elastic loading, or unloading, a hypoelastic formulation of Hooke's law was relied upon. As already stated above, kinematic hardening effects were not included in the analysis. Such effects can certainly influence the outcome of a scratch test but particularly so during the unloading sequence of the test. In this analysis, the loading part of the scratch test is of primary interest and for this reason, and also for clarity (the interpretation of the results becomes more involved due to an increased number of constitutive parameters), only isotropic hardening is considered here.

As for the delamination analysis, it is well known that the delamination process can be severely influenced by mode-mixity, and for this reason, a general criterion was used according to

G = [G.sub.c] ([psi]), (4)

as suggested by Hutchinson and Suo (33) for mixed-mode crack propagation along a weak plane. This energy may then be allowed to vary depending on the mode of crack growth (i.e., mode I, mode II, or a combination thereof). The mode mixity may be described by a parameter [PSI], (33) defined as

[psi] = (2/[pi])arctan([K.sub.II]/[K.sub.I]). (5)

In equation (5), [K.sub.I] and [K.sub.II] are mode I and II stress intensity factors, respectively. In this study, this dependence was investigated through the use of the double cantilever beam (DCB) loaded by uneven bending moments (UBM) as described by Sorensen and Jacobsen (34) and Sorensen et al. (35) According to these investigations, the energy release rate may be calculated using the J-integral as

G = J = ((21([([M.sub.1]).sup.2] + [([M.sub.2]).sup.2]) - 6[M.sub.1][M.sub.2]) / (4[B.sup.2][H.sup.2]E)) x (1 - [[upsilon].sup.2]), (6)

assuming plane deformation and |[M.sub.1]| < [M.sub.2] (see Fig. 4). In equation (6), B is the specimen width, H is the specimen height, and E and [upsilon] are the elastic constants. In this case, equation (5) can also be expressed in terms of the bending moments (35) (Fig. 4) acting on the DCB as

[psi] = (2/[pi])arctan(([square root of (3)][M.sub.1] + [M.sub.2] / (2[M.sub.1] - [M.sub.2])), |[M.sub.1]| < [M.sub.2], [7]

[FIGURE 4 OMITTED]

The DCB-UBM was chosen since it exhibits the attractive quality of the energy release rate being independent of the crack location (as the crack length is not present in equation 6), making it only necessary to measure the bending moments during the testing. In order to be able to model the creation of new surface with the accompanying separation energy at delamination of a film/substrate interface numerically, a cohesive zone model was implemented. Normally, the cohesive laws are defined through an interfacial potential [phi] with a traction vector T = ([T.sub.n], [T.sub.t]) acting on the cohesive surface (cf. Xu and Needleman (36) Needleman (37) and Ortiz and Pandolfi (38) as

T = [[[partial derivative][phi]([DELTA])]/[[partial derivative][DELTA]]] (8)

where [DELTA] = ([[DELTA].sub.n], [[DELTA].sub.t]) indicate the separation of the surfaces. In general, as the cohesive surfaces separate, the traction will increase until a maximum is reached after which it will decrease to zero resulting in complete separation (see Fig. 5). Consequently, the area under the curve is the energy needed for separation.

[FIGURE 5 OMITTED]

In this analysis, as indicated above, it was assumed that the material was adequately described by either classical elastoplasticity or the standard Drucker-Prager (32) material model describing pressure-sensitive flow. Despite this, however, the resulting boundary value problem became very involved (in particular so, as a film/substrate boundary was introduced into the problem) and it was necessary to use the finite element method to arrive at results of acceptable accuracy. This numerical analysis was here performed with the aid of the commercial FEM package ABAQUS. (39) As for details of the numerical analysis, we refer to Wredenberg and Larsson (16) and Larsson and Wredenberg (24) for monolithic and film/substrate systems, and only some basic features and details concerning the cohesive zone analysis will be discussed below. As already stated above, regarding the constitutive specification, the incremental rate independent Drucker-Prager (32) equations for pressure-sensitive large deformation with isotropic hardening, were relied upon.

Considering that the material experienced very large strains, adaptive meshing was used to maintain the element integrity. The mesh (shown in Fig. 6) was composed of some 80,000 eight-node linear-reduced integration elements. These elements were chosen since they show a faster convergence with respect to mesh refinement than tetrahedral elements and do not have the inherent contact problems of quadratic elements. (39) The stylus was assumed to be perfectly rigid, and Coulomb friction was assumed when appropriate. All of the results of this study are pertinent to a perfectly sharp conical stylus. In order to insure a relevant comparison between experimental and numerical results, experimental scratch depths were chosen in such a way that any influence from the stylus tip defect was negligible (as investigated carefully in each case through numerical simulations). As regards boundary conditions, the surface outside the contact area was assumed traction free and within the area of contact unilateral kinematic constraints, given by the shape of the rigid conical indenter/stylus depicted in Fig. 1, was imposed.

[FIGURE 6 OMITTED]

To account for the adhesive bond between the film and the (rigid) substrate, 10,000 cohesive elements with a traction-separation law were implemented (see Fig. 7). The cohesive law used was defined by five parameters: the maximum cohesive stresses:[[sigma].sub.max,n] and [[sigma].sub.max,t], and the critical energy release rates: [G.sub.c]([psi] = 0), [G.sub.c]([psi] [approximately equal to] 0.45), [G.sub.c]([psi] = 1). The damage initiation criterion (the dashed tine in Fig. 7) could be expressed as

[([[sigma].sub.n]/[[sigma].sub.max,n]).sup.2] + [([[sigma].sub.1]/[[sigma].sub.max,t]).sup.2] = 1, (9)

[FIGURE 7 OMITTED]

where [[sigma].sub.n] and [[sigma].sub.t] are the tractions in the normal and shear direction, respectively.

The cohesive elements were for numerical purposes given an initial stiffness of 30 TPa/m corresponding to the initial slope of the cohesive law. A study of the importance of this parameter on the delamination load was performed, and it was concluded that any of the tested stiffness values ranging from 20 to 80 TPa/m would suffice (the results differed with less than 0.5%). Ultimately, the stiffness of 30 TPa/m was chosen as it allows reasonably large elements, and thus lower computational times, as a stiffer cohesive law requires a denser element mesh.

Delamination was said to occur when some cohesive element experienced more than 99% of accumulative damage, meaning that the cohesive deformation had passed the point of maximum traction, and that the tractions were down to 99% of the maximum tractions of an undamaged element (refer to Fig. 5). This choice was not of any real importance, but was chosen to avoid any numerical peculiarities involved in the case of a value being 100% accumulated damage. Indeed, changing this value to 100% would cause an indistinguishable change in the simulated delamination load.

Finally, in the context of theory and numerics, it should be emphasized once again that the scheme presented here is essentially the same as the one presented in a parallel article by Wredenberg and Larsson, (28) but in this investigation, only the influence from different mechanical and geometrical quantities at classical elastoplasticity was studied.

Experimental analysis

For the present analysis, experiments related to scratching and fracture mechanics were conducted. All the experiments were performed at room temperature. Again, it should be mentioned that essentially the same experimental scheme and results were presented in the parallel article by Wredenberg and Larsson. (28)

The scratch setup

Scratching experiments were performed by dragging a conical diamond (with the angle [beta] = 22[degrees]) over the surface of a specimen, using an MTS 66202A-01 biaxial servo-hydraulic machine with an Instron 8500 control unit. The normal forces were measured with an MTS load cell, and the tangential forces were measured by a load cell built in-house. During the test, the normal and tangential forces were recorded. The rotational motion of the actuator was transformed to a translational motion by a carriage sliding on a rail. During scratching, the normal load was linearly increased. After the scratching procedure, the specimen was removed, and the point of delamination was recorded and related to the corresponding load. The delamination load was found by measuring the point of initial delamination on the scratched specimen as shown in Fig. 8. As the load was linearly increased during scratching, the delamination load could be found through linear interpolation of the applied scratch load.

[FIGURE 8 OMITTED]

Scratch speeds of 1 and 0.1 mm/s over a total scratch length of 42 mm were used. The experimental results proved to be somewhat dependent on scratch speed (to be discussed in some detail below). Scratching was initialized at a normal load of 50 N and linearly increased to 150 N during the procedure.

Determination of cohesive parameters

To determine the mix-mode behavior of the cohesive law, double cantilever beam specimens were prepared. The specimens were subsequently loaded with uneven bending moments to achieve an arbitrary mode-mixity, [psi], (35) as can be seen in Fig. 4. The contribution of the film strip on the J-integral was assumed to be small as the stiffness of the steel part of the DCB was much larger and the film strip was thin compared to the steel, thus allowing the energy release rate to be calculated according to equation (10). As an example, if the contribution of the film strips were taken into account for the case of [M.sub.1] = 0, this would result in a J approximately 2% larger. The maximum stresses ([[sigma].sub.max,n] and [[sigma].sub.max,t] as discussed in some detail earlier, see Fig. 7) in the cohesive law were determined by uniaxial tensile and pure shear tests, where the film was glued between two cylindrical steel bars. The bars were then either pulled apart (tensile) or twisted (shear) with increasing force until the film/steel interface failed. The shear lest was set up in the same configuration as the DCB test, with pure bending moments as to minimize the normal stresses introduced.

Preparation of specimens

Each of the scratch specimens was composed of a steel substrate with a Vinyl-Ester film adhered to it by means of an epoxy resin. It should be noted that any influence from the mechanical behavior of the epoxy was taken into account indirectly through the cohesive law. Prior to adhesion to the substrate, the film was machined to a thickness of 0.9 mm from a thick slate of which bulk mechanical properties had already been investigated by Wredenberg and Larsson. (17) The film had a surface roughness with a standard deviation of approximately 40 nm and keeping in mind that the scratch depth during the experiments was around 10 [micro]m or higher, this roughness should have negligible influence on relevant test results.

After the film was placed on top of the substrate, the specimens were vacuum bagged at 50 kPa and allowed to cure for 24 h at 40[degrees]C. When completed, each specimen was approximately 25 mm wide by 55 mm long and 15 mm thick. The resulting adhesive layer was approximately 6 [micro]m thick.

The DCB specimens were prepared in a similar fashion. Here, two steel beams were adhered to a film strip by use of the same epoxy resin as used for the scratch specimen (see Fig. 4). To allow for a well-defined initial crack the film strip (and also the steel beams) was given a coat of wax covering the first two centimeters. These specimens were also vacuum bagged and allowed to cure in the same way as in the case of the scratch specimens. Here, the resulting adhesive layer was approximately 8 [micro]m thick.

The cylindrical specimens were pushed together by springs (as the shape of the specimen was not suitable for vacuum bagging), giving an adhesive thickness of approximately 5 [micro]m after being cured for 24 h at 40[degrees]C. In all specimens the film thickness was 0.9 mm.

Results and discussion

Initially in this section, the relevant experimental results (scratching and cohesive zone results) are presented and discussed. Then, the experimental results are compared with corresponding numerical ones. Such a comparison is certainly of substantial interest in itself but is presently also of interest to insure the reliability of the numerical approach to, in particular, delamination at scratching of thin film/substrate systems. This feature is of direct importance in the present investigation as regards the accuracy of the results from the numerical study of pressure-sensitive flow effects on relevant scratch quantities. Of course, the results pertinent to this numerical study constitute the most important outcome of this article.

Starting then by discussing the experimental results, the normal load at which delamination of the Vinyl-Ester film from the substrate occurred, [F.sub.n,delam], can be seen in Table 1, along with the normal scratch hardness, [H.sub.n] (equation 1), at the two different scratch speeds. The hardness was evaluated at half of the scratch length, i.e., the scratch width was measured right in the middle between the starting point and finishing point of the residual groove to avoid any influence from the specimen boundary. The experimentally determined projected normal contact area, [A.sub.norm], was calculated assuming a contact of semi circular shape as

[A.sub.norm] = [(w/2).sup.2] ([pi]/2), (10)
Table 1: Experimental results for the normal delamination load,
[F.sub.n,delam], and normal scratch hardness [H.sub.n] (standard
deviation)

Scratch speed (mm/s)  [F.sub.n,delam] (N)  [H.sub.n ]([MP.sub.a])

0.1                       97.5 (8.3)              367 (20)
1.0                       110 (11.1)              365 (10)


according to standard procedure at scratch testing (cf. Williams (40)). In equation (10) w is the residual groove width. As can be seen in Table 1, the delamination load data did not show a significant dependence of the scratch speed and furthermore, the mode of failure did not in anyway change due to scratch speed. In this context, it should also be mentioned that all scratches experienced some cracking in the wake of the stylus. The load at which these cracks formed varied from sample to sample. In around one third of the scratches, delamination occurred prior to the formation of such cracks. When investigating this phenomenon, no correlation between the formation of cracks and the delamination load could be found. It should be noted that any analytical expression, or for that matter any attempt to determine an analytical expression, for the delamination load, pertinent to the present problem, has not been found in the literature. This was not attempted here either remembering that the problem is notoriously nonlinear due to large deformations and contact related effects. Accordingly, only a straightforward comparison between corresponding numerical and experimental results is relied upon here.

As regards the model parameters to be introduced in the numerical analysis, this mainly concerned the constitutive parameters pertinent to the Vinyl-Ester film as well as the interfacial coefficient of friction [[micro].sub.i]. As mentioned above, the Vinyl-Ester material has been experimentally characterized previously by Wredenberg and Larsson, (17) and it suffices here only to repeat those findings. The experimentally determined uniaxial stress-strain (effective plastic strain [[epsilon].sub.p]) curve for the Vinyl-Ester film material is shown in Fig. 9. Obviously, this material exhibits softening after initial plastic deformation. Softening is then followed by extensive hardening, which could not be easily described constitutively as cracking then occurred during the tensile test (as well as barreling during a corresponding compressive test). This issue is discussed somewhat extensively below. Furthermore, combined with additional experiments for thin films by Wredenberg and Larsson, (28) the interfacial coefficient of friction [[micro].sub.i] was determined to be approximately 0.07 (the general behavior of most of the global quantities discussed in this analysis is quantitatively much influenced by the presence of interfacial friction, in contrast to the situation at normal indentation, cf. e.g., Carlsson et al. (41)). Young's modulus was measured to be 3.5 GPa. In the present analysis also, the material constants pertinent to pressure-sensitive flow, modeled by a Drucker-Prager (32) constitutive law, is of interest. Such results were not published by Wredenberg and Larsson (17) but from their experimental results, it is possible to determine the "friction angle" in equation (3) as [alpha] = 2[degrees].

[FIGURE 9 OMITTED]

As regards the fracture mechanics results, DCB experiments were performed for three DBM ratios, [M.sub.1]/[M.sub.2] = -1, 0, 1, corresponding to mode mixities [psi] = 0, [psi] [approximately equal to] 0.45, and [psi] = 1 (see equation 7). As can be seen in Fig. 10, the critical energy release rate is substantially higher for pure mode II ([psi] = 1) than for pure mode I ([psi] = 0) loading. The maximum normal cohesive stress and maximum shear cohesive stress were found to be [[sigma].sub.max,n] = 21 MPa and [[sigma].sub.max,t] = 27 MPa, respectively.

[FIGURE 10 OMITTED]

In order to determine the reliability of the numerical approach, a comparison between numerical and experimental results pertinent to Vinyl-Ester was performed. Accordingly, corresponding values for delamination load and (normal) scratch hardness were compared. It was then assumed that quasi-static conditions prevailed, i.e., the uniaxial stress-strain relations were determined from experiments performed at such values on the strain rate that rate effects can be neglected. In this context, it should be mentioned that some small rate effects were noticed when the scratch experiments were performed at different speeds, see Table 1. As the comparison aimed at quasi-static conditions, the experimental results pertinent to a scratch speed being 0.1 mm/s were used for this purpose. The shape of the stress-strain curve used in the numerical model deserves some further discussion. As previously mentioned, the material showed softening after initial deformation, and then hardening according to Fig. 9. This indicates that the present value of the well-known Johnson (14) contact parameter

[lambda] = (E tan [beta]) / ([[sigma].sub.rep](1 - [[upsilon].sup.2])) (11)

was approximately 21 ([[sigma].sup.rep] is normally a representative stress value, and is also here, equal to the value of the uniaxial stress at a plastic strain [[epsilon].sub.p] = 0.08). However, as also discussed above, the subsequent hardening could not be well described due to cracking (and barreling) during the uniaxial tensile test. For this reason, the stress-strain behavior was modeled, in the numerical simulations, as, first of all, a material described by no hardening after softening (here called material A) and also a material which behaves perfectly plastic after initial yield (here called material B). However, in case of material B, the actual elastic modulus is adjusted in such a way that the A-value, equation (11), was 21 as determined from the uniaxial experiments. For material A, of course, the elastic modulus is 3.5 GPa as reported from the experiments discussed above.

Remembering that both softening and hardening characterized the material, it seems likely that these effects are fairly well represented in average by material B, and for this reason, this material was singled out to be used in the numerical calculations when closeness to experimental conditions was to be investigated. For simplicity, classical Mises plasticity was assumed in these numerical calculations. A comparison between experimental and numerical results is shown in Table 2, where it should be emphasized that the hardness values reported correspond to the hardness for a monolithic Vinyl-Ester material. In this case, the numerical scheme is identical to the one presented by Wredenberg and Larsson (16), (17) and is not discussed further here (the finite element mesh, however, is shown in Fig. 11 with Cartesian coordinates indicated for clarity). Obviously, the two sets of results are close, and remembering the difficulties involved in interpreting the outcome of the scratch/delamination experiments, this feature gives some definite confidence in the numerical procedure used in the present analysis. In particular so, remembering that first of all qualitative effects are of importance as regards the parametric study (even though quantitative results also are of interest). It should be noted in passing that also normal cone indentation of Vinyl-Ester was performed experimentally and numerically (material B), and in this case the two sets of results were almost identical which gives further confidence in the numerical procedure.

[FIGURE 11 OMITTED]
Table 2: Comparison of simulated and experimental results for the
normal delamination load, [F.sub.n,delam], and the normal scratch
hardness, [H.sub.n] (standard deviation)

              [F.sub.n,delam] (N)  [H.sub.n ]([MP.sub.a])

Experimental  97.5 (8.3)           367 (20)
Simulated     88                   344


As a summary, then concerning the constitutive behavior used in the numerical analysis, material B (perfectly plastic after initial yield) was used for the above comparison between numerical and experimental results. The reason for this was that this material represents, on average, the softening followed by hardening behavior found in the numerical experiments. It was thought advisable not to model the hardening part of the experiments explicitly due to the fact that cracking (and barreling) then disturbed the interpretation of the experiments. However, as the present investigation is devoted toward investigating pressure-sensitive flow effects in polymers, material A is a natural choice for reference material in the following part of the numerical analysis due to the fact that pressure sensitivity then can be studied on a material with softening after initial plastic deformation, a common feature when polymers are at issue. It should be emphasized that in this part of the investigation, closeness to experimental results is not of immediate importance.

Having thus shown the accuracy of the present numerical approach, it now seems advisable to start the presentation and discussion of results pertinent to scratching of materials exhibiting pressure-sensitive flow. This is done by first analyzing the results for the two hardness quantities defined in equations (1) and (2). In this case, scratching of monolithic materials are considered (as previously mentioned the finite element mesh used in these calculations is shown in Fig. 11) to avoid any effects from the boundary, i.e., the film/substrate interface.

Accordingly then, in Figs. 12 (normal hardness) and 13 (tangential hardness), the two hardness quantities are depicted as functions of the friction angle [alpha] in equation (3). The material properties are pertinent to material A but, of course, with a varying value on the friction angle. It is obvious from the results in Figs. 12 and 13 that the hardness values are quite dependent of [alpha]. Initially, however, this is not so and accordingly, pressure-sensitive flow is not a major issue (at least not for the hardness values) in the case of the presently tested Vinyl-Ester material. As a result, the comparison in Table 2 is accurate and valid for this polymer. At higher values of the friction angle, the numerical results start to deviate from classical elastoplasticity ([alpha] = 0), and at [alpha] = 8[degrees], this difference is well above 10%. From an experimental point of view, this is outside the normal scatter at indentation/scratch testing, and accordingly this feature has to be accounted for when interpreting experimental results. Only small differences of results were found between the normal and tangential hardness quantities.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

The influence from pressure-sensitive flow on the delamination behavior at scratching will now be discussed. Obviously then, thin film scratching as outlined above is investigated numerically with material A as a reference material but with a varying value on the friction angle. It should first be mentioned that the delamination of the Vinyl-Ester film was shear driven, as (when performing numerical experiments) increasing the cohesive strength in the normal mode had no effect on the delamination load, whereas setting the cohesive strength in the shear mode to infinity led to an "un-delaminable" interface. A typical delamination shape is shown in Fig. 14 where the initiation of failure always occurred at the interface slightly in front of the indenter tip but within a region defined by the projection of the contact area on the interface.

[FIGURE 14 OMITTED]

The delamination behavior in the context of applied scratch load at the occurrence of initial delamination is shown in Figs. 15 and 16. The results for the normal load component, [F.sub.n,delam], are shown in Fig. 15 and the corresponding tangential load, [F.sub.t,delam], in Fig. 16 (corresponding preliminary results have been presented in reference 28). It is obvious that the delamination load is, in contrast to the situation concerning the hardness values, only marginally dependent on the pressure sensitivity. Indeed, the only effect of interest is that at [alpha] = 8[degrees] the load starts to increase somewhat. The behavior is the same for both the normal and the tangential delamination load and from an experimental point of view, the change of the delamination load as function of friction angle is within the normal experimental scatter. Accordingly, any effects from pressure sensitivity as regards this issue can not be determined for the Vinyl-Ester film investigated experimentally here.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

A possible physical explanation of the fact that hardness values are more sensitive to Drucker-Prager plasticity features than the delamination load can be found from results by Larsson and Wredenberg. (24) In this article, among other things, the behavior of the stress field at monolithic and thin film scratching of a classical elastoplastic material is presented (Figs. 7 and 11 in reference 24). It is shown that, for example, the first principal stress component is mainly compressive but on the surface near the stylus high tensile stresses can also be found. Accordingly, the difference in tensile and compressive yield stress due to pressure sensitivity is more noticeable at the surface of the thin film, and as a consequence, the contact pressure is more dependent on the friction angle [alpha] than the shear stresses at the interface.

The depth at which delamination occurred was small. This indicates that the global scratch properties such as hardness, etc., are virtually unaffected by the presence of the substrate according to previous study by Larsson and Wredenberg. (24) This analysis also provides further details as regards the influence of coating thickness and in combination with the results by Wredenberg and Larsson, (25) where the effect from a deformable substrate is investigated, features related to coating thickness and properties variation are much better known. Explicit results from the present investigation also indicates that even when h/d = 0.25 substrate effects on the delamination load are very small (in all the calculations, delamination initialed at values on h/d being much smaller than 0.25).

It should be emphasized that the present choice of attack on the delamination problem, using cohesive zone modeling, was made based on the fact that such an analysis does not require a pre-existing crack. As an alternative, a direct attack on the delamination problem based on linear fracture mechanics, in the spirit of for example Nilsson et al. (42) or Larsson, (43) is, of course, also a possible but less accurate alternative. In this context, it should be mentioned that, for example, Martin et al. (44) have emphasized that for a general analysis of delamination problems both fracture stress and fracture energy must be considered. Obviously, the present analysis confirms this conclusion.

As a final comment, a short discussion seems advisable about the fact that the numerical analysis above is restricted to values on the friction angle [alpha] between 0[degrees] and 8[degrees]. Basically, the reason for this restriction is the numerical difficulties occurring at higher values on [alpha]. However, it is obvious that the present results are pertinent to many polymers of practical interest (for the Vinyl-Ester material investigated here [alpha] = 2[degrees]) and that they also indicate clear trends for higher [alpha]-values.

Conclusions

This investigation was performed in parallel with the study by Wredenberg and Larsson. (28) In reference (28), a comprehensive study of thin film scratching of classical elastoplastic materials was performed with particular emphasis on crack initiation and growth. In this study, the influence on the scratch problem from pressure-sensitive flow was studied in some detail.

In summary, it can be concluded that pressure sensitivity has to be accounted for when high accuracy is at issue, since global scratch properties such as hardness were much dependent on this feature. However, it should also be mentioned that the influence from pressure-sensitive flow on the delamination load proved to be small and well within any experimental scatter.

Acknowledgment The authors would like to acknowledge the financial support received through grant #621-2005-5803 from the Swedish Research Council.

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F. Wredenberg, P.-L. Larsson (*)

Department of Solid Mechanics, Royal Institute of Technology, 10044 Stockholm, Sweden

e-mail: pelle@hallf.kth.se

Present Address:

F. Wredenberg

Scania CV, SE-151 87, Sodertalje, Sweden

J.Coat. Technol. Res., 7(3) 279-290, 2010

DOI 10.1007/s11998-009-9202-4
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