# On the stability of delamination growth at scratching of thin film structures.

Abstract Scratching of thin film/substrate structures is studied
theoretically

and numerically. The results are discussed in connection to delamination initiation and in particular subsequent growth at scratching. The material behavior of the film is described by classical elastoplasticity accounting for large deformations. The deformation of the substrate is neglected indicating that the results are pertinent to soft thin films. The numerical investigation is performed using the finite element method (FEM) and the numerical strategy is discussed in some detail. The results from this study show that delamination growth at thin film scratching is a stable feature with crack arrest occurring at a decreasing load.

Keywords Scratching, Film/substrate structures, Delamination growth, Stability, FEM analysis

Introduction

The quantities obtained during an indentation test may be used to determine the constitutive properties of the tested materials, with the early theoretical works by Sneddon,1 Tabor, and Johnson3 being the most notable for this purpose. The commonly determined properties using indentation are hardness and the indentation load versus indentation depth curve. The indentation hardness represents a measure of the average pressure under the indenter tip and is calculated from the force necessary to push a rigid indenter into the material. An additional important parameter involved in the hardness calculation is the actual area of contact between the indenter and the material, usually estimated using suitable assumptions.

Depending on the material properties and the type of indenter used, Johnson3 suggested that the outcome of a sharp indentation test on classical elastoplastic materials could be placed in one of three levels as specified by the parameter

[LAMBDA]= {EPSILON] tan [beta]/((l- [[nu].sup.2])[[delta].sub.rep])

In equation (1), {EPSILON] is the Young's modulus and [nu] is Poisson's ratio, [beta] is the angle between the (sharp) indenter and the undeformed surface and [[delta].sub.rep] is the material flow stress at a representative value of the effective (accumulated) plastic strain[[epsilon].sub.p] chosen below to be the [[epsilon].sub.p] = 0.08. As for the three indentation levels, schematically shown in Fig. 1, level I, A < 3, corresponds to the occurrence of very little plastic deformation during indentation testing, meaning that all global properties can be derived from an elastic analysis. In level II, 3 < A < 30, an increasing amount of plastic deformation is present and both the elastic and plastic properties of the material will influence the outcome of the test, and finally, in level III, A > 30, plastic deformation is present over the entire contact area. The last mentioned level is applicable to most engineering metals. From a number of tests performed on different materials pertinent to level III, [Tabor.sup.2] concluded that a simple formula relating hardness, flow stress and a constant dependent on the geometry of the sharp indenter could be derived. Although Tabor's method is basically empirical, since the complex stress distribution at elastoplastic indentation rules out an analytically based relation, it is known to give good accuracy in a number of cases.

It is a well-known fact that microindentation provides information about elastic and plastic deformation on a localized scale and it is particularly attractive for thin films with a typical thickness of a few micrometers or less. As much as indentation testing has been used, few theoretical methods are 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 and numerical methods like the finite element method (FEM) are necessary tools for determining relevant quantities in an accurate manner. The early works by Bhattacharya and Nix,4'5 Laursen and Simo,6 and Giannakopoulos and coworkers7-10 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 (11-14) and, perhaps more generally, in reference (15). It should be noted in passing that many of these FEM studies of thin film indentation were inspired by the development of the nanoindenter (Pethica et al.).(16)

The fundamental knowledge about the mechanical behavior at scratching, which currently is also a well established technique for hardness testing, is not nearly as developed as for indentation testing. Some early mechanical analyses concerning different aspects of scratching of metals should be mentioned though. (17-20)

It goes almost without saying that due to the complexity of the boundary value problem, the analysis of this type of hardness testing also requires numerical methods; again FEM is to be preferred, for a high accuracy analysis. In recent years, a number of such analyses have been presented, cf., e.g., references (21-23) with the most important conclusion being that the Johnson parameter in equation (1) can also be used to correlate scratching experiments.

Also, thin film scratching has been studied quite frequently, cf., e.g., references (24-34) considering different aspects of the problem such as fracture, delamination, the behavior of different global and local scratch variables (global scratch variables include normal and tangential hardness, apparent coefficient of friction and contact area) and alternative constitutive behavior. In reference (33), the local stress fields at thin film scratching were examined in some detail and an important conclusion from this study was that high shear stresses are the dominant factor leading to delamination initiation and growth. However, in this context some preliminary results, briefly discussed in reference (33), indicated that delamination growth is a stable process and crack arrest will occur at constant or decreasing load. This feature was for brevity not investigated further in reference (33) but is indeed very important as it suggests that damage at scratching can be contained in a practical situation. The stability of delamination growth at scratching has not been investigated in full previously and it is the present intention to remedy this shortcoming. In doing so, the numerical strategy, based on the FEM, developed by Wredenberg and Larsson (22),(23) was used while modified to also account for film/substrate boundary effects. For clarity and convenience, but not out of necessity, the analysis was restricted to cone scratching ([beta]= 22[degrees] in Fig. 2) of elastoplastic materials, as in such a case the indenter does not introduce any characteristic length in the problem. In this context, it should be mentioned that [beta]= 22[degrees] was chosen since this is a representative value for most sharp indenter geometries of practical interest. Furthermore, only soft homogeneous coatings are considered and accordingly, the deformation of the substrate is neglected in the analysis. The present results can of course then not be extended to coating systems with hard/nonhomogeneous coatings. However, in case of hard or, more correctly, harder coatings some guidance is given in reference (32) as regards the validity of the assumption of a stiff substrate.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Basic considerations and numerical analysis

In the present analysis scratching of thin film/substrate systems using a sharp rigid conical stylus is studied. In the presentation below h is the scratch depth and d is the film thickness, as shown in Fig. 2 and A is the normally projected contact area between stylus and material. Quasi-static and steady-state conditions are assumed to prevail and accordingly, the monolithic scratch problem is self-similar with no characteristic length present. Consequently, the normal and tangential scratch hardness as well as the ratio h/ [square root of (term)]A are constant during the loading sequence of a scratch test on homogeneous materials and stresses and strains are functions of the dimensionless variables [x.sub.i]/ [square root of (term)]A (the Cartesian coordinate system is shown in Fig. 3) 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 hid, d being the film thickness as shown in Fig. 2. However, when the indentation depth is held constant during the test, steady-state conditions prevail and the global quantities discussed above are constant, facilitating a direct comparison with the corresponding homogeneous solution when warranted.

In the present analysis, as mentioned above, it is assumed that the material behavior is adequately described by classical elastoplasticity. Despite this, though, the resulting boundary value problem becomes very involved (in particular so as a film/substrate boundary is introduced into the problem) and it is necessary to use the FEM in order to arrive at results of acceptable accuracy. The basic numerical scheme for an analysis of the corresponding homogeneous problem was developed by Wredenberg and [Larsson.sup.22,23] and further improved for thin film problems by Larsson and Wredenberg. (30) This scheme will be closely adhered to also in the present investigation and consequently, below, only the most important elements in this numerical approach are discussed with the particular ingredients due to the presence of a film/ substrate system described in some detail. Furthermore, the additional numerical implications due to delamination crack initiation and subsequent growth will be discussed in the next section.

[FIGURE 3 OMITTED]

For the constitutive specification the incremental, rate independent Prandtl-Reuss equations for classical large deformation von Mises plasticity with isotropic hardening were implemented. At elastic loading, or unloading, a hypoelastic formulation of Hooke's law, pertinent to the elastic part of the Prandtl-Reuss equations, was relied on. Obviously, within this setting, 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 the present 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. For the particular situation analyzed here, a soft film on a hard substrate, polymeric films are the most natural class of materials to analyze. In doing so, a vinyl-ester material, exhibiting both softening and hardening after initial yield, was singled out for attention. The material has been previously characterized by Wredenberg and [Larsson.sup.31,34] and the material properties for this material are

[pounds sterling] = 3.5GPa, v = 0.5, [LAMBDA] = 21, (2)

with the behavior after initial yielding shown in Fig. 4. It should also be mentioned that, as pointed out by Wredenberg and Larsson, (31),(34) as the hardening region in Fig. 4 could not be well described due to cracking and barrelling during uniaxial testing, this feature was not included in the numerical simulations.

The numerical simulations were performed using FEMs implemented in the commercial FEM package ABAQUS. (35) As the material experienced very large strains, arbitrary lagrangian eulerian (ALE) adaptive meshing was used to maintain the element integrity. In the case of thin film scratching, the mesh was composed of approximately 80,000 linear eight-noded elements and is shown in Fig. 5 (the corresponding mesh at scratching of monolithic (homogeneous) materials is not shown here for brevity but is presented by, for example, Wredenberg and [Larsson.sup.23]). The elements were of hybrid type, i.e., the displacements were approximated with bilinear shape functions while the hydrostatic pressure attained a constant value in each element, in order to improve convergence at almost incompressible deformation. Linear elements were chosen instead of quadratic elements as they do not show the inherent contact problems of such elements, ABAQUS. (35) It should also be emphasized that the number of elements used in the numerical simulations were not fixed but varied somewhat so that the contact area included as many elements as possible in order to ensure high accuracy results. As regards boundary conditions, the surface outside the contact area was assumed traction free and within the region of contact unilateral kinematic constraints, given by the shape of the rigid, conical indenter were accounted for. In this context, it should be mentioned that loading was applied by prescribing the normal and tangential displacement of the rigid stylus. In order to simulate the film/substrate boundary zero displacements and rotations were imposed along the lower surface (not shown) of the mesh in Fig. 5. Obviously, as already mentioned above, this corresponds to the case of a soft film on a hard (stiff) substrate with perfect bonding (except of course at cracking) along the interface. At monolithic scratching the outer boundaries of the mesh were chosen to be sufficiently far away from the contact area in order to avoid any remote boundary effects. This corresponds to scratching of a homogeneous half-space. Finally, it should be mentioned that frictional effect was for completeness included in the analysis of the polymeric film discussed above. In doing so, standard Coulomb friction was assumed with a coefficient of friction

[FIGURE 4 OMITTED]

[mu] = 0.07, (3)

cf., e.g., Wredenberg and Larsson, (31) representative of this type of polymer. It should be noted in this context that both local and global scratch properties are very much dependent on frictional effects. This is in contrast to the situation at normal indentation as then at least global parameters such as hardness are almost independent of friction, cf., e.g., Carlsson et al.(36)

[FIGURE 5 OMITTED]

Delamination analysis

In a series of recent investigations, Wredenberg and Larsson. (31), (34) used cohesive zone analysis to study delamination initiation and growth at scratching. Further developments regarding this matter were presented by Wredenberg and Larsson. (33) In reference (33), the details concerning the stress fields at thin film scratching were analyzed and it was concluded that high shear stresses were the driving force for both crack initiation and continued growth along the film/substrate interface. In this investigation, the load at crack forming and subsequent growth is not of immediate interest but instead, the stability of delamination growth in such a situation is studied. Consequently, only the most important features of a numerical analysis of delamination at thin film scratching are presented here directly below. For more details we refer to Wredenberg and Larsson.(31),(34)

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 energy release rate-based criterion was used according to

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

as suggested by Hutchinson and Suo37 for mixed mode crack propagation along a weak plane. In equation (4), [G.sub.c] is the separation energy necessary for the delamination of the film. 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), Kx and Ku are mode I and II stress intensity factors, respectively. Furthermore, to be able to model the creation of a new surface with the accompanying separation energy at delamination of a film/substrate interface numerically, a cohesive zone model was implemented. Commonly, the cohesive laws are defined through an interfacial potential [empty set] with a traction vector T = ([T.sub.n],[T.sup.t]) acting on the cohesive surface (cfM Xu and Needleman, (38) Needleman, (39) and Ortiz and [Pandolfi.sup.40]) as

T = - a[empty set]([DELTA])/a[DELTA] (6)

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. 6). Consequently, the area under the curve is the energy needed for separation.

In the numerical analysis, the adhesive bond between the film and the (rigid) substrate was accounted for using 10,000 cohesive elements with an implemented traction-separation law as shown in Fig. 7. The cohesive law used was defined by five parameters: the maximum cohesive stresses ([[delta].sub.max,n]and[[delta].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][approximately equal to]1)). The damage initiation criterion (the dashed line in Fig. 7) could be expressed as

[([delta].sub.n/[delta].sub.max,n]).sup.2] + [([delta].sub.t/[delta].sub.max,t]).sup.2] = 1, (7)

where an and ert are the tractions in the normal and shear direction, respectively. It should be emphasized that the explicit values on the five parameters were determined experimentally for the vinyl-ester polymer film adhered to a steel substrate at issue by Wredenberg and Larsson31"34 and are also used as a representative material here. Explicit values for the critical energy release rates are shown in Fig. 8 while the cohesive stresses take on the values [[delta].sub.max,n]= 21 MPa and [[delta].sub.max,t]= 27 MPa, 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 from 20 to 80 TPa/m would suffice (the results differed with <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.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

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 1% of the maximum tractions of an undamaged element (refer to Fig. 6). This choice was not of any real importance, but was chosen in order 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.

[FIGURE 9 OMITTED]

Finally, in this context it should be emphasized once again that a determination of such features as for example delamination load and shape was not of immediate interest in the present analysis. Consequently, the explicit values on the cohesive parameters presented above, and used in the numerical simulations, are not of direct importance but should be regarded as being representative for a typical soft polymeric coating.

Results and discussion

As an introduction to the present discussion it seems advisable to show some results by Wredenberg and Larsson, (33)already discussed briefly above, pertinent to the present case. Accordingly then, stress field trajectories for scratching of the polymeric film/substrate system, with no delamination present, are shown in Fig. 9 (normal stresses) and in Fig. 10 (shear stress). It is obvious that the normal stress acting as a possible driving force for the initiation of a delamination, [[delta].sub.33], is compressive at possible failure sites along the interface. The maximum principal stress attains some tensile values in this region but since it is clear from Fig. 9 that the direction of this stress does not coincide with the interface, normal, tensile stresses will play a secondary role as regards the initiation of delamination cracks in this region. However, the shear stresses at the interface are high. Indeed, they attain their highest values at the film/substrate interface. Accordingly, as mentioned above, based on the results in Figs. 9 and 10 it was then concluded by Wredenberg and [Larsson.sup.33] that shear at the film/substrate interface is the driving force for the initiation of delamination. It should be noted that it was shown in reference (33) that this is the case also for metallic films and for polymeric films with pressure-sensitive flow accounted for. For this reason, and for brevity, alternative constitutive descriptions will not be included in the present analysis.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

In order to achieve further understanding of the delamination process in combination with the related stress fields it is also important to investigate the situation at the progressing crack growth. This is the main feature of interest presently. In doing so, a small delamination (the delaminated area is schematically indicated in Fig. 11), and a larger one (indicated in Fig. 13) is investigated. It should be noted that in this case the delamination size is related to a representative length of the contact region, for example, [square root of (term)]A.

The normal stress [[delta].sub.33]and the shear stress [t.sup.23] in the case of a small delamination are shown in Fig. 12. It can be observed that the maximum values of the shear stress are no longer present along the interface but can be found inside the film. In comparison with the stresses presented in Fig. 10 the values at the interface in front of or below the delamination are noticeably lower. This can in part be explained by the fact that delamination occurs at relatively small values on the indentation depth, i.e., hid < 0.28 (being the indentation depth in Figs. 9 and 10), but the main reason for this is the creation of new surfaces at delamination which obviously leads to the relaxation of shear stresses. As regards the normal stress [[delta].sub.33] it remains compressive along the interface with explicit values very close to the ones reported in Fig. 9. For brevity, the first principal stress is not shown.

[FIGURE 13 OMITTED]

At increasing load the delaminated area will continue to grow as shown in Fig. 13. The geometry of the delamination is more elongated, compared to the small almost circular geometry in Fig. 11. This could be expected though as the growth of the damaged area occurs under increasing load and this matter is not dwelled upon any further.

The stress trajectories, corresponding to the large delamination in Fig. 13, are presented in Fig. 14 and should be compared to the ones in Fig. 12, pertinent to a small delamination. Clearly, the difference between the two sets of results is small. Indeed, as regards shear stresses slightly lower values are observed in Fig. 14 which is surprising as the applied load (acting on the stylus) is larger in this case. Furthermore, as was the case also in Fig. 12, high shear stresses are only found inside the film and at the interface these stresses are very low. Again, the normal stress [[delta].sub.33]remains compressive and does not change much at all during the growth process (this conclusion is only relevant when stress values are reported in a coordinate system following the conical stylus as behind the indenter tip unloading occurs).

The results presented in Figs. 9-14 can now be summarized with the stability of crack growth particularly in mind. First of all, it is clear that, as already suggested by Wredenberg and Larsson, (33) the main driving force behind the initiation of a delamination crack is high shear stresses at the film/ substrate interface. However, when a delamination is formed the region of high shear stresses immediately moves away from the interface and is then situated inside the thin film. Furthermore, delamination growth decreases the explicit values of the shear stresses of interest. Normal stresses at the interface are compressive, and remain so during crack growth, and can accordingly not balance the reduction in driving force due to decreasing shear stresses. Furthermore, it has been shown in previous investigations by Wredenberg and Larsson, (33)-(34) that this fundamental mechanical behavior is essentially not influenced by such features as film thickness, substrate deformation, and pressure sensitive flow. In summary, this leads to the conclusion that delamina-tion growth at scratching is a stable process and unless the scratch loads (depth) are dramatically increased at the instance of initiation, such failure will not be catastrophic (here defined as continued crack growth at constant or decreasing outer loading). Having said this, though, it should be mentioned that the analysis is restricted to quasi-static conditions and dynamic effects imposed from high loading rates (scratch speed) are not accounted for and are left for future investigations.

It seems appropriate to emphasize that the present analysis is restricted to classical elastoplasticity with material properties pertinent to a polymeric film. However, results presented in reference (33) concerning the stress distribution at thin film scratching indicate a similar behavior also for metallic films and for polymeric films with pressure-sensitive flow accounted for. It could be argued then that the conclusions above are relevant also for such types of material behavior.

As a final comment, it should be emphasized once again that in this study, crack initiation and growth was introduced into the problem by using cohesive zone analysis. It deserves to be mentioned that a direct attack on the delamination problem based on linear fracture mechanics, in the spirit of, for example, Nilsson et al. (41) or Larsson, (42) is of course also a possible but indeed a less accurate alternative as it requires some assumption about the shape of the initial delamination.

[FIGURE 14 OMITTED]

Conclusions

Scratching of thin film/substrate systems was analyzed numerically in order to determine the stability of delamination growth in such a situation. The analysis is restricted to the case of a soft layer on a rigid substrate. The results presented indicate that delamination growth at scratching is a stable process and unless the scratch loads (depth) are dramatically increased at the instance of crack initiation, such failure will not be catastrophic. The investigation is restricted to polymeric films modeled by standard elastoplasticity but based on the stress field characteristics presented in previous investigations for a similar situation, it could be argued that this conclusion applies also for other types of material behavior.

Acknowledgments The authors want to acknowledge the support through grant 621-2005-5803 from the Swedish Research Council.

References

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Oxford, 1951

F. Wredenberg, P.-L. Larsson (El) Department of Solid Mechanics, Royal Institute of Technology, 10044 Stockholm, Sweden e-mail: pelle@hallf.kth.se

F. Wredenberg

Swerea Kimab AB. 10216 Stockholm, Sweden

DOI 10.1007/sll 998-011-9352-z

and numerically. The results are discussed in connection to delamination initiation and in particular subsequent growth at scratching. The material behavior of the film is described by classical elastoplasticity accounting for large deformations. The deformation of the substrate is neglected indicating that the results are pertinent to soft thin films. The numerical investigation is performed using the finite element method (FEM) and the numerical strategy is discussed in some detail. The results from this study show that delamination growth at thin film scratching is a stable feature with crack arrest occurring at a decreasing load.

Keywords Scratching, Film/substrate structures, Delamination growth, Stability, FEM analysis

Introduction

The quantities obtained during an indentation test may be used to determine the constitutive properties of the tested materials, with the early theoretical works by Sneddon,1 Tabor, and Johnson3 being the most notable for this purpose. The commonly determined properties using indentation are hardness and the indentation load versus indentation depth curve. The indentation hardness represents a measure of the average pressure under the indenter tip and is calculated from the force necessary to push a rigid indenter into the material. An additional important parameter involved in the hardness calculation is the actual area of contact between the indenter and the material, usually estimated using suitable assumptions.

Depending on the material properties and the type of indenter used, Johnson3 suggested that the outcome of a sharp indentation test on classical elastoplastic materials could be placed in one of three levels as specified by the parameter

[LAMBDA]= {EPSILON] tan [beta]/((l- [[nu].sup.2])[[delta].sub.rep])

In equation (1), {EPSILON] is the Young's modulus and [nu] is Poisson's ratio, [beta] is the angle between the (sharp) indenter and the undeformed surface and [[delta].sub.rep] is the material flow stress at a representative value of the effective (accumulated) plastic strain[[epsilon].sub.p] chosen below to be the [[epsilon].sub.p] = 0.08. As for the three indentation levels, schematically shown in Fig. 1, level I, A < 3, corresponds to the occurrence of very little plastic deformation during indentation testing, meaning that all global properties can be derived from an elastic analysis. In level II, 3 < A < 30, an increasing amount of plastic deformation is present and both the elastic and plastic properties of the material will influence the outcome of the test, and finally, in level III, A > 30, plastic deformation is present over the entire contact area. The last mentioned level is applicable to most engineering metals. From a number of tests performed on different materials pertinent to level III, [Tabor.sup.2] concluded that a simple formula relating hardness, flow stress and a constant dependent on the geometry of the sharp indenter could be derived. Although Tabor's method is basically empirical, since the complex stress distribution at elastoplastic indentation rules out an analytically based relation, it is known to give good accuracy in a number of cases.

It is a well-known fact that microindentation provides information about elastic and plastic deformation on a localized scale and it is particularly attractive for thin films with a typical thickness of a few micrometers or less. As much as indentation testing has been used, few theoretical methods are 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 and numerical methods like the finite element method (FEM) are necessary tools for determining relevant quantities in an accurate manner. The early works by Bhattacharya and Nix,4'5 Laursen and Simo,6 and Giannakopoulos and coworkers7-10 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 (11-14) and, perhaps more generally, in reference (15). It should be noted in passing that many of these FEM studies of thin film indentation were inspired by the development of the nanoindenter (Pethica et al.).(16)

The fundamental knowledge about the mechanical behavior at scratching, which currently is also a well established technique for hardness testing, is not nearly as developed as for indentation testing. Some early mechanical analyses concerning different aspects of scratching of metals should be mentioned though. (17-20)

It goes almost without saying that due to the complexity of the boundary value problem, the analysis of this type of hardness testing also requires numerical methods; again FEM is to be preferred, for a high accuracy analysis. In recent years, a number of such analyses have been presented, cf., e.g., references (21-23) with the most important conclusion being that the Johnson parameter in equation (1) can also be used to correlate scratching experiments.

Also, thin film scratching has been studied quite frequently, cf., e.g., references (24-34) considering different aspects of the problem such as fracture, delamination, the behavior of different global and local scratch variables (global scratch variables include normal and tangential hardness, apparent coefficient of friction and contact area) and alternative constitutive behavior. In reference (33), the local stress fields at thin film scratching were examined in some detail and an important conclusion from this study was that high shear stresses are the dominant factor leading to delamination initiation and growth. However, in this context some preliminary results, briefly discussed in reference (33), indicated that delamination growth is a stable process and crack arrest will occur at constant or decreasing load. This feature was for brevity not investigated further in reference (33) but is indeed very important as it suggests that damage at scratching can be contained in a practical situation. The stability of delamination growth at scratching has not been investigated in full previously and it is the present intention to remedy this shortcoming. In doing so, the numerical strategy, based on the FEM, developed by Wredenberg and Larsson (22),(23) was used while modified to also account for film/substrate boundary effects. For clarity and convenience, but not out of necessity, the analysis was restricted to cone scratching ([beta]= 22[degrees] in Fig. 2) of elastoplastic materials, as in such a case the indenter does not introduce any characteristic length in the problem. In this context, it should be mentioned that [beta]= 22[degrees] was chosen since this is a representative value for most sharp indenter geometries of practical interest. Furthermore, only soft homogeneous coatings are considered and accordingly, the deformation of the substrate is neglected in the analysis. The present results can of course then not be extended to coating systems with hard/nonhomogeneous coatings. However, in case of hard or, more correctly, harder coatings some guidance is given in reference (32) as regards the validity of the assumption of a stiff substrate.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Basic considerations and numerical analysis

In the present analysis scratching of thin film/substrate systems using a sharp rigid conical stylus is studied. In the presentation below h is the scratch depth and d is the film thickness, as shown in Fig. 2 and A is the normally projected contact area between stylus and material. Quasi-static and steady-state conditions are assumed to prevail and accordingly, the monolithic scratch problem is self-similar with no characteristic length present. Consequently, the normal and tangential scratch hardness as well as the ratio h/ [square root of (term)]A are constant during the loading sequence of a scratch test on homogeneous materials and stresses and strains are functions of the dimensionless variables [x.sub.i]/ [square root of (term)]A (the Cartesian coordinate system is shown in Fig. 3) 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 hid, d being the film thickness as shown in Fig. 2. However, when the indentation depth is held constant during the test, steady-state conditions prevail and the global quantities discussed above are constant, facilitating a direct comparison with the corresponding homogeneous solution when warranted.

In the present analysis, as mentioned above, it is assumed that the material behavior is adequately described by classical elastoplasticity. Despite this, though, the resulting boundary value problem becomes very involved (in particular so as a film/substrate boundary is introduced into the problem) and it is necessary to use the FEM in order to arrive at results of acceptable accuracy. The basic numerical scheme for an analysis of the corresponding homogeneous problem was developed by Wredenberg and [Larsson.sup.22,23] and further improved for thin film problems by Larsson and Wredenberg. (30) This scheme will be closely adhered to also in the present investigation and consequently, below, only the most important elements in this numerical approach are discussed with the particular ingredients due to the presence of a film/ substrate system described in some detail. Furthermore, the additional numerical implications due to delamination crack initiation and subsequent growth will be discussed in the next section.

[FIGURE 3 OMITTED]

For the constitutive specification the incremental, rate independent Prandtl-Reuss equations for classical large deformation von Mises plasticity with isotropic hardening were implemented. At elastic loading, or unloading, a hypoelastic formulation of Hooke's law, pertinent to the elastic part of the Prandtl-Reuss equations, was relied on. Obviously, within this setting, 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 the present 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. For the particular situation analyzed here, a soft film on a hard substrate, polymeric films are the most natural class of materials to analyze. In doing so, a vinyl-ester material, exhibiting both softening and hardening after initial yield, was singled out for attention. The material has been previously characterized by Wredenberg and [Larsson.sup.31,34] and the material properties for this material are

[pounds sterling] = 3.5GPa, v = 0.5, [LAMBDA] = 21, (2)

with the behavior after initial yielding shown in Fig. 4. It should also be mentioned that, as pointed out by Wredenberg and Larsson, (31),(34) as the hardening region in Fig. 4 could not be well described due to cracking and barrelling during uniaxial testing, this feature was not included in the numerical simulations.

The numerical simulations were performed using FEMs implemented in the commercial FEM package ABAQUS. (35) As the material experienced very large strains, arbitrary lagrangian eulerian (ALE) adaptive meshing was used to maintain the element integrity. In the case of thin film scratching, the mesh was composed of approximately 80,000 linear eight-noded elements and is shown in Fig. 5 (the corresponding mesh at scratching of monolithic (homogeneous) materials is not shown here for brevity but is presented by, for example, Wredenberg and [Larsson.sup.23]). The elements were of hybrid type, i.e., the displacements were approximated with bilinear shape functions while the hydrostatic pressure attained a constant value in each element, in order to improve convergence at almost incompressible deformation. Linear elements were chosen instead of quadratic elements as they do not show the inherent contact problems of such elements, ABAQUS. (35) It should also be emphasized that the number of elements used in the numerical simulations were not fixed but varied somewhat so that the contact area included as many elements as possible in order to ensure high accuracy results. As regards boundary conditions, the surface outside the contact area was assumed traction free and within the region of contact unilateral kinematic constraints, given by the shape of the rigid, conical indenter were accounted for. In this context, it should be mentioned that loading was applied by prescribing the normal and tangential displacement of the rigid stylus. In order to simulate the film/substrate boundary zero displacements and rotations were imposed along the lower surface (not shown) of the mesh in Fig. 5. Obviously, as already mentioned above, this corresponds to the case of a soft film on a hard (stiff) substrate with perfect bonding (except of course at cracking) along the interface. At monolithic scratching the outer boundaries of the mesh were chosen to be sufficiently far away from the contact area in order to avoid any remote boundary effects. This corresponds to scratching of a homogeneous half-space. Finally, it should be mentioned that frictional effect was for completeness included in the analysis of the polymeric film discussed above. In doing so, standard Coulomb friction was assumed with a coefficient of friction

[FIGURE 4 OMITTED]

[mu] = 0.07, (3)

cf., e.g., Wredenberg and Larsson, (31) representative of this type of polymer. It should be noted in this context that both local and global scratch properties are very much dependent on frictional effects. This is in contrast to the situation at normal indentation as then at least global parameters such as hardness are almost independent of friction, cf., e.g., Carlsson et al.(36)

[FIGURE 5 OMITTED]

Delamination analysis

In a series of recent investigations, Wredenberg and Larsson. (31), (34) used cohesive zone analysis to study delamination initiation and growth at scratching. Further developments regarding this matter were presented by Wredenberg and Larsson. (33) In reference (33), the details concerning the stress fields at thin film scratching were analyzed and it was concluded that high shear stresses were the driving force for both crack initiation and continued growth along the film/substrate interface. In this investigation, the load at crack forming and subsequent growth is not of immediate interest but instead, the stability of delamination growth in such a situation is studied. Consequently, only the most important features of a numerical analysis of delamination at thin film scratching are presented here directly below. For more details we refer to Wredenberg and Larsson.(31),(34)

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 energy release rate-based criterion was used according to

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

as suggested by Hutchinson and Suo37 for mixed mode crack propagation along a weak plane. In equation (4), [G.sub.c] is the separation energy necessary for the delamination of the film. 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), Kx and Ku are mode I and II stress intensity factors, respectively. Furthermore, to be able to model the creation of a new surface with the accompanying separation energy at delamination of a film/substrate interface numerically, a cohesive zone model was implemented. Commonly, the cohesive laws are defined through an interfacial potential [empty set] with a traction vector T = ([T.sub.n],[T.sup.t]) acting on the cohesive surface (cfM Xu and Needleman, (38) Needleman, (39) and Ortiz and [Pandolfi.sup.40]) as

T = - a[empty set]([DELTA])/a[DELTA] (6)

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. 6). Consequently, the area under the curve is the energy needed for separation.

In the numerical analysis, the adhesive bond between the film and the (rigid) substrate was accounted for using 10,000 cohesive elements with an implemented traction-separation law as shown in Fig. 7. The cohesive law used was defined by five parameters: the maximum cohesive stresses ([[delta].sub.max,n]and[[delta].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][approximately equal to]1)). The damage initiation criterion (the dashed line in Fig. 7) could be expressed as

[([delta].sub.n/[delta].sub.max,n]).sup.2] + [([delta].sub.t/[delta].sub.max,t]).sup.2] = 1, (7)

where an and ert are the tractions in the normal and shear direction, respectively. It should be emphasized that the explicit values on the five parameters were determined experimentally for the vinyl-ester polymer film adhered to a steel substrate at issue by Wredenberg and Larsson31"34 and are also used as a representative material here. Explicit values for the critical energy release rates are shown in Fig. 8 while the cohesive stresses take on the values [[delta].sub.max,n]= 21 MPa and [[delta].sub.max,t]= 27 MPa, 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 from 20 to 80 TPa/m would suffice (the results differed with <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.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

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 1% of the maximum tractions of an undamaged element (refer to Fig. 6). This choice was not of any real importance, but was chosen in order 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.

[FIGURE 9 OMITTED]

Finally, in this context it should be emphasized once again that a determination of such features as for example delamination load and shape was not of immediate interest in the present analysis. Consequently, the explicit values on the cohesive parameters presented above, and used in the numerical simulations, are not of direct importance but should be regarded as being representative for a typical soft polymeric coating.

Results and discussion

As an introduction to the present discussion it seems advisable to show some results by Wredenberg and Larsson, (33)already discussed briefly above, pertinent to the present case. Accordingly then, stress field trajectories for scratching of the polymeric film/substrate system, with no delamination present, are shown in Fig. 9 (normal stresses) and in Fig. 10 (shear stress). It is obvious that the normal stress acting as a possible driving force for the initiation of a delamination, [[delta].sub.33], is compressive at possible failure sites along the interface. The maximum principal stress attains some tensile values in this region but since it is clear from Fig. 9 that the direction of this stress does not coincide with the interface, normal, tensile stresses will play a secondary role as regards the initiation of delamination cracks in this region. However, the shear stresses at the interface are high. Indeed, they attain their highest values at the film/substrate interface. Accordingly, as mentioned above, based on the results in Figs. 9 and 10 it was then concluded by Wredenberg and [Larsson.sup.33] that shear at the film/substrate interface is the driving force for the initiation of delamination. It should be noted that it was shown in reference (33) that this is the case also for metallic films and for polymeric films with pressure-sensitive flow accounted for. For this reason, and for brevity, alternative constitutive descriptions will not be included in the present analysis.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

In order to achieve further understanding of the delamination process in combination with the related stress fields it is also important to investigate the situation at the progressing crack growth. This is the main feature of interest presently. In doing so, a small delamination (the delaminated area is schematically indicated in Fig. 11), and a larger one (indicated in Fig. 13) is investigated. It should be noted that in this case the delamination size is related to a representative length of the contact region, for example, [square root of (term)]A.

The normal stress [[delta].sub.33]and the shear stress [t.sup.23] in the case of a small delamination are shown in Fig. 12. It can be observed that the maximum values of the shear stress are no longer present along the interface but can be found inside the film. In comparison with the stresses presented in Fig. 10 the values at the interface in front of or below the delamination are noticeably lower. This can in part be explained by the fact that delamination occurs at relatively small values on the indentation depth, i.e., hid < 0.28 (being the indentation depth in Figs. 9 and 10), but the main reason for this is the creation of new surfaces at delamination which obviously leads to the relaxation of shear stresses. As regards the normal stress [[delta].sub.33] it remains compressive along the interface with explicit values very close to the ones reported in Fig. 9. For brevity, the first principal stress is not shown.

[FIGURE 13 OMITTED]

At increasing load the delaminated area will continue to grow as shown in Fig. 13. The geometry of the delamination is more elongated, compared to the small almost circular geometry in Fig. 11. This could be expected though as the growth of the damaged area occurs under increasing load and this matter is not dwelled upon any further.

The stress trajectories, corresponding to the large delamination in Fig. 13, are presented in Fig. 14 and should be compared to the ones in Fig. 12, pertinent to a small delamination. Clearly, the difference between the two sets of results is small. Indeed, as regards shear stresses slightly lower values are observed in Fig. 14 which is surprising as the applied load (acting on the stylus) is larger in this case. Furthermore, as was the case also in Fig. 12, high shear stresses are only found inside the film and at the interface these stresses are very low. Again, the normal stress [[delta].sub.33]remains compressive and does not change much at all during the growth process (this conclusion is only relevant when stress values are reported in a coordinate system following the conical stylus as behind the indenter tip unloading occurs).

The results presented in Figs. 9-14 can now be summarized with the stability of crack growth particularly in mind. First of all, it is clear that, as already suggested by Wredenberg and Larsson, (33) the main driving force behind the initiation of a delamination crack is high shear stresses at the film/ substrate interface. However, when a delamination is formed the region of high shear stresses immediately moves away from the interface and is then situated inside the thin film. Furthermore, delamination growth decreases the explicit values of the shear stresses of interest. Normal stresses at the interface are compressive, and remain so during crack growth, and can accordingly not balance the reduction in driving force due to decreasing shear stresses. Furthermore, it has been shown in previous investigations by Wredenberg and Larsson, (33)-(34) that this fundamental mechanical behavior is essentially not influenced by such features as film thickness, substrate deformation, and pressure sensitive flow. In summary, this leads to the conclusion that delamina-tion growth at scratching is a stable process and unless the scratch loads (depth) are dramatically increased at the instance of initiation, such failure will not be catastrophic (here defined as continued crack growth at constant or decreasing outer loading). Having said this, though, it should be mentioned that the analysis is restricted to quasi-static conditions and dynamic effects imposed from high loading rates (scratch speed) are not accounted for and are left for future investigations.

It seems appropriate to emphasize that the present analysis is restricted to classical elastoplasticity with material properties pertinent to a polymeric film. However, results presented in reference (33) concerning the stress distribution at thin film scratching indicate a similar behavior also for metallic films and for polymeric films with pressure-sensitive flow accounted for. It could be argued then that the conclusions above are relevant also for such types of material behavior.

As a final comment, it should be emphasized once again that in this study, crack initiation and growth was introduced into the problem by using cohesive zone analysis. It deserves to be mentioned that a direct attack on the delamination problem based on linear fracture mechanics, in the spirit of, for example, Nilsson et al. (41) or Larsson, (42) is of course also a possible but indeed a less accurate alternative as it requires some assumption about the shape of the initial delamination.

[FIGURE 14 OMITTED]

Conclusions

Scratching of thin film/substrate systems was analyzed numerically in order to determine the stability of delamination growth in such a situation. The analysis is restricted to the case of a soft layer on a rigid substrate. The results presented indicate that delamination growth at scratching is a stable process and unless the scratch loads (depth) are dramatically increased at the instance of crack initiation, such failure will not be catastrophic. The investigation is restricted to polymeric films modeled by standard elastoplasticity but based on the stress field characteristics presented in previous investigations for a similar situation, it could be argued that this conclusion applies also for other types of material behavior.

Acknowledgments The authors want to acknowledge the support through grant 621-2005-5803 from the Swedish Research Council.

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Oxford, 1951

F. Wredenberg, P.-L. Larsson (El) Department of Solid Mechanics, Royal Institute of Technology, 10044 Stockholm, Sweden e-mail: pelle@hallf.kth.se

F. Wredenberg

Swerea Kimab AB. 10216 Stockholm, Sweden

DOI 10.1007/sll 998-011-9352-z

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Author: | Wredenberg, Fredrik; Larsson, Per-Lennart |
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Publication: | JCT Research |

Article Type: | Report |

Date: | Nov 1, 2011 |

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