On material characterization of paper coating materials by microindentation testing.
Keywords: Non-destructive testing, microindentation, mechanical properties, packaging, paper, FEM
In order to improve the printability and the optical properties of paper and board, a thin coating layer is often applied. This is especially so in modern digital printing operations where the paper is, almost without exception, coated in order to provide better printability. However, coating the paper can also lead to new problems not encountered in uncoated paper. For example, cracking of the coating is very common during folding of the printed paper. (1) In order to understand how the coating influences various paper properties, a mechanical analysis is often imperative. In such an analysis, coated paper can be regarded as a three-layered composite material or a sandwich material composed of a paper sheet as a middle layer with porous outer layers of a highly mineral-filled latex polymer on both sides (see Figure 1). A typical thickness for paper used in digital printing is 0.10-0.15 mm, with the total thickness of the coating accounting for approximately 10%. Due to the fact that the coating is very thin, material characterization becomes an issue of some difficulty during a mechanical investigation of coated papers. This is especially so as uniaxial tensile testing is not a simple and reliable alternative for these types of materials. (2) A widely used method is the determination of the shear modulus with a torsion pendulum, or torsional braid, where glass fibers are impregnated with the material to be tested and dynamical measurements are done. (3) As an alternative to traditional testing, indentation or hardness testing is frequently used for material characterization. In particular, when it comes to the characterization of thin films or coatings, the advantages of indentation are obvious as discussed in numerous articles in the literature, (2-6) just to mention a few, and it is the intention here to utilize this experimental method in order to determine relevant material properties of paper coating materials.
The most important quantities given by an indentation test are the hardness, H (mean contact pressure at maximum loading), the contact area, A, and the relation between indentation load, P, and indentation depth, h, during loading as well as unloading. These parameters can then be used in order to determine the constitutive behavior of the material by taking advantage of results from previous investigations by, for example, Hertz, (7) Tabor, (8) Johnson, (9-10) Doerner and Nix, (11) and Oliver and Pharr. (12) In short, these investigations present closed-form relations between indentation quantities and material properties to be used at material characterization. Johnson (9-10) also showed that the outcome of an indentation test (on classical elastic-plastic materials) will fall into one of three different levels, depending on the material behavior and the type of indenter used. These three levels can be characterized as follows. In level I, very little plastic deformation occurs during indentation and all global properties can be derived from an elastic analysis. In level II, an increasing amount of plastic deformation is present and both the elastic and plastic properties of the material will influence the indentation parameters determined during the test. Finally, then, in level III, plastic deformation is present all over the contact area and elasticity no longer influences global indentation quantities. In this context, it should be emphasized that during a sharp indentation test (Berkovich, Vickers, Knoop, or cone indenters) plastic deformation is present almost without exception and such tests will fall either into level II or level III. This is not so, however, for spherical indenters (or other blunt indenters) as then the presence of a characteristic length leads to a situation where relevant indentation quantities are varying during a single test. Accordingly, at spherical indentation, the test will fall into level I initially, then enter level II, and subsequently level III at progressing loading. This feature is to be taken advantage of in the present investigation, as will be discussed in much more detail.
[FIGURE 1 OMITTED]
The investigations, as well as the results, discussed above are mainly pertinent to material characterization of bulk materials by indentation. As previously mentioned though, indentation has in recent years been used frequently for determining material properties of thin films and coatings, and this is now perhaps the main area of application for this particular experimental method. There are certainly many practical difficulties that arise when performing experiments at the small length scales involved at thin film indentation (in most applications the thickness of the film is approximately 1-10 [micro]m), but perhaps the most important problem concerns the influence from the film/substrate interface (and from the substrate itself) on relevant indentation properties. In short, due to this feature, well-known formulae for material characterization by indentation are no longer sufficiently accurate, cf. e.g., Cai and Bangert (13) and Mitre Peterson and Larsson. (14) An immediate remedy to this problem is, of course, to use nanoindentation in order to perform extremely small indentations. This will, however, increase the cost, as well as the experimental complexity, of the procedure considerably and it seems desirable to, if possible, avoid such an approach here.
[FIGURE 2 OMITTED]
The intention of the present study was to investigate if indentation can be used as a reliable and simple tool in order to determine the elastic properties of highly mineral-filled latex polymers used as paper coating materials. The determination of the plastic properties are certainly also of substantial interest, but as this constitutes a more involved problem it was thought advisable to limit this introductory study to linear elastic material behavior. As a consequence, a spherical indenter is most appropriate to use as then, remembering the previous discussion, advantage can be taken of the initial elastic regime present at spherical indentation (level I). Furthermore, it has already been stated that nanoindentation is not an option in this investigation, remembering that simplicity is an important variable. However, in case of microindentation (here interpreted as small scale indentation performed by using standard experimental devices), high experimental accuracy can only be expected for indentation depths larger than 0.1-0.5 [micro]m. With film thicknesses ranging from 5-10 [micro]m in the present applications, it is very hard to avoid substantial influence from the film/substrate interface on important indentation quantities. In addition, the paper substrate (paper sheet) itself will deform during a straightforward indentation experiment on a coated paper and, consequently, a correct interpretation of the experimental results is even more difficult to achieve. A remedy to these problems would be to perform indentation experiments on single films built up to a thickness substantially larger than the one encountered in a real situation. From a practical point of view, it is certainly possible to produce the necessary (thicker) specimens and to perform the microindentation experiments, but it remains to be determined if the material characteristics of the (relatively) thick specimens are close to the corresponding behavior of the thin paper coatings. In order to avoid any confusion, it should be mentioned that using thick specimens is of great advantage at indentation but does not substantially improve the situation at uniaxial testing due to gripping problems and microcracking of the films.
It should be noted that the present approach laid down above is not at all obvious and there are other related studies in the literature where alternative methods, excluding nanoindentation experiments, have been suggested for determining the bulk properties of thin films or coatings from indentation experiments. Bhattacharya and Nix (15) used the finite element method in order to relate the hardness at cone indentation of film/substrate systems to the constitutive properties of the two materials involved. The resulting empirical formulae were, however, quite involved and do not apply to the more complicated case of spherical indentation. It should also be remembered that the substrate material in the present application is paper, which is a highly anisotropic material. Accordingly, for the reasons discussed, any attempt to correlate the constitutive behavior of the polymeric coating and the paper sheet into a universal relation describing relevant indentation quantities appears to be an overwhelming task and was not carried through here.
In summary then, the present study concerns the determination of the elastic material properties of highly mineral-filled latex polymers, often used as paper coating materials, using microindentation testing with spherical indenters. The experiments will be performed on films that are substantially thicker than the ones used in a real situation. The paper sheet itself is not under investigation presently and, accordingly, during the experiments the thin films will be placed on a stiff metal foundation. The indentation results will be compared with corresponding results from very carefully conducted uniaxial tensile tests. Finite element calculations are to be used in order to determine the influence from the finite film thickness and from plastic effects. Finally, indentation as a tool for also determining the failure stress of the polymer films will be discussed in some detail.
DESCRIPTION OF THE PROBLEM: THEORETICAL BACKGROUND
Based on the discussion above, it is the intention of the present analysis to aim towards a simple and reliable experimental method for determining the (elastic) material properties of highly mineral-filled latex polymers, and the method of choice is spherical indentation testing. Initially, when interpreting the experimental results, it is tentatively assumed that the polymer films are sufficiently thick (and that the indentation depths are small enough) so that the geometrical boundaries of the film do not influence relevant indentation properties. The validity of this assumption will be discussed below and investigated in detail by presently performed finite element calculations. The influence from plastic deformation will also be adressed in the same manner.
The classical contact theory developed by Hertz (7) constitutes a solid foundation for the presently proposed experimental approach. The original work by Hertz (7) concerned frictionless contact between two deformable elastic spheres, but the results are also directly applicable to the present case of a (comparatively) rigid metal spherical indenter and a deformable elastic polymer half-space, (see Figure 2). For such a situation, Hertz (7) showed that the indentation load, P, could be expressed as
P = [4[Ea.sup.3]]/[3R(1 - [v.sup.2])] (1)
where a is the contact radius indicated in Figure 2, E and v are the modulus of elasticity and Poisson's ratio of the polymer film, respectively, and R is the radius of the spherical indenter. At Hertzian contact, the contact radius and the indentation depth, h, are related according to
[a.sup.2]/Rh = 1. (2)
By combining (1) and (2), the formula
P = [4E[h.sup.3/2][square root of R]]/[3(1 - [v.sup.2])] (3)
results. Consequently, at spherical indentation of an elastic half-space the indentation load, P, is proportional to [h.sup.3/2], and this feature will be taken advantage of when analyzing the results from the indentation experiments.
It should be emphasized once again that the theory developed by Hertz (7) rests on the assumption that the deformable body is so large that the outer boundaries do not influence the resulting field variables. Normally, as a rule of thumb, it is assumed that when indentation depths are smaller than approximately 1/10 of the film thickness, boundary effects are negligible. It was shown, however, by Mitre Peterson and Larsson (14) that this is a nonconservative estimate in many cases and particularly at elastic indentation; being of most immediate interest presently, boundary effects are substantial at much smaller values on the indentation depth. This issue will be investigated in detail using the finite element method as outlined below.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Another feature that can decrease the predictive capability of equations (1-3) is the influence from plasticity. It was stated earlier that purely elastic conditions are found only during the very initial stages of a spherical indentation test (level I). Johnson (9-10) showed that the variable
[LAMBDA] = Ea/[(1 - [v.sup.2])[[sigma].sub.y]R'] (4)
where [[sigma].sub.y] is the initial yield stress of the material, could be a useful quantitative tool in order to separate the different levels of indentation (level I-III) as discussed. Based on theoretical and experimental results, Johnson (9-10) also suggested that [LAMBDA] < 3 was sufficient in order to ensure elastic conditions. It should be immediately emphasized that [LAMBDA] < 3 can only be considered as an approximate guideline for the present experimental setup, as this parameter is also influenced by the strain-hardening characteristics of the material. In such a case, the initial yield stress [[sigma].sub.y] should be replaced by the flow stress at a representative value of the plastic strain, [[epsilon].sub.r] = 0.2(a/R), as suggested by Johnson. (9-10) The coating materials used in the present investigation were clay and calcium carbonate. Uniaxial stress-strain curves for these materials were determined experimentally, as will be discussed, but as their strain-hardening characteristics cannot be easily determined at high strain levels, equation (4) will be used as an approximative tool for designing the experimental setup.
[FIGURE 5 OMITTED]
Finally, in this context, it should be mentioned that viscoelastic effects were not accounted for in this study. Admittedly, time-dependent deformation is a common feature for many coating materials but could be neglected presently, based on experimental findings discussed below, without introducing any significant errors in the interpretation of the experimental results. However, viscoelastic effects could be introduced in the analysis without too much effort following the theoretical and experimental findings by Larsson and Carlsson. (16)
With the theoretical foundation for the present analysis now properly outlined, it seems appropriate to go into some detail with regards to the experimental setup. It has already been stated that it is not possible to perform indentation tests directly on the coated paper, as then the paper sheet will unavoidably influence the indentation quantities. Instead, the indentation experiments were performed on thin layers of coating materials placed on a rigid metal foundation. In a real situation the coatings would be, as mentioned, approximately 1-10 [micro]m thick. This would require high accuracy results for indentation depths around 0.1 [micro]m or less in order to secure a sufficient number of data points with indentation results that are not influenced by the lower boundary of the coating--and this is not possible to achieve by using standard experimental devices. The surface roughness of the coating would also influence the results at such levels of indentation depth. Accordingly, indentation was performed on coating layers being 30 [micro]m and 70 [micro]m thick, as this would allow for the evaluation of results for indentation depths of the order of 1 [micro]m or even larger. Furthermore, with such values of the maximum indentation depths and with material constants according to material 2 in Table 1 (expected to be close to the actual material characteristics of the coating materials), it was found based on Johnson's (9-10) suggestion [[LAMBDA] < 3 in equation (4)] that a sphere diameter of at least 1 cm was needed in order to avoid substantial influence from plasticity. Accordingly, in the experiments two spheres with diameters of 1 cm and 3 cm were used.
The experimental setup is shown in some detail in Figure 3. Loading was applied using a hydraulic testing machine (MTS Systems Corp.) with the rate of indentation displacement being approximately 0.001 mm/sec [experiments were performed with different values on the rate of indentation depth (0.001, 0.002, and 0.008 mm/sec) and no noticeable difference of results was found, indicating that viscoelastic effects were negligible for these materials]. The displacement of the indenter was measured and controlled continuously using two induction gages while the load was recorded with a load cell. The tests were performed at room temperature under controlled conditions. The spherical indenter and the metal foundation were both made of steel with a modulus of elasticity being equal to 206 GPa. As the corresponding values for the film materials at issue were, approximately, 5 GPa or even less, the deformation of the indenter will be negligible and assumed rigid in the analysis. The reliability of the experimental results were tested by performing microindentation experiments on the steel foundation in the elastic region (stage I). In short, perfect agreement, with regards to indentation load and indentation depth, was found between the experimental results and theoretical ones based on Hertz (7) classical theory (in this case, the deformation of both the indenter and the foundation have to be accounted for), which certainly gives some confidence to the presently used experimental setup.
The coating materials used in this study were, as previously mentioned, highly filled polymers with clay or calcium carbonate as pigment. Both materials were produced in thin films on a base substrate, and subsequently separated from the substrate, and in two versions with 8% and 16% (percentage of the mineral weight) of the same latex binder with a glass transition temperature ([T.sub.g]) of 13[degrees]C. A review of different techniques on how to produce and separate coating films has been presented by Prall. (17) Presently, wet coating was applied to a Mylar film using a rod coater (R K Print) and the coating was dried in an oven at 105[degrees]C for three minutes. Due to the low stiffness of the Mylar film, as well as the low adhesion to the coating, it is possible to peal it off without severe cracking of the coating. Thickness measurements after separation indicated that the coatings were uniform in terms of thickness and that they, accordingly, could be used for both indentation testing and uniaxial tensile testing. Using this technique, clay coatings with the target thicknesses (30 [micro]m and 70 [micro]m) were produced while only 70-[micro]m thick calcium carbonate films were successfully separated from the Mylar films (calcium carbonate films with lower thicknesses failed at separation).
During drying of the coatings, water evaporates in only one direction and this might affect the distribution in the thickness direction of the mineral particles and of the latex binder. For this reason, it is not at all obvious that the films produced are homogeneous in composition and that, subsequently, the mechanical behavior of the thick layers of the materials is the same as the corresponding behavior of the thin coatings used in a practical situation. In order to find at least some guidance in this matter, a microscopy investigation of the thin coatings with a film thickness of 70 [micro]m was conducted. Some representative results from this investigation, performed using environmental scanning electron microscope (ESEM), are shown in Figure 4, and judging from these pictures, the composition of the films are almost surprisingly uniform with regards to the distribution of the mineral particles. Accordingly, it seems justified to assume that the mechanical behavior of the thick coatings closely resembles the behavior of the same material when used in a coated paper. It should also be mentioned that the microscopy investigation showed a difference in the shape of the mineral particles--the clay particles were like small plates while the calcium carbonate particles were more spherical.
[FIGURE 6 OMITTED]
In this study, explicit values determined from the uniaxial tensile tests were considered to be the best approximation of the elastic stiffness of the materials investigated. Judging from the previous discussion, this is not an obvious assumption but was based on the fact that great efforts were made in order to reduce the cracking of the thin films at production and to avoid gripping problems at load application. The specimens used for testing were made by cutting the coatings into strips of dimensions (15 mm X 150 mm). This was done prior to the separation from the Mylar films. The tests were conducted using an Alwetron TH-1 (Lorentzen & Wettre) tensile tester with a strain rate approximately corresponding to the one used in the indentation experiments. A clamping length of 100 mm was used and the film thicknesses were approximately 70 [micro]m. For each set of coating material, 10 strips were tested and representative results are shown in Figure 5. Explicitly, the average values on the elastic stiffnesses determined from the uniaxial tests were (assuming v = 0.3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Clearly, no distinction has been made between results for different weight percent of latex, as corresponding values for these materials were very close (Figure 5), and well within the limits of the experimental accuracy. It should be noted in passing that the elastic stiffness for clay is higher than the corresponding one for calcium carbonate, as reported previously by Lepoutre and Rigdahl, (18) based on uniaxial tensile tests.
A typical P-h curve given by the indentation experiments is shown in Figure 6. Depending on the thickness of the film, the experimental results were evaluated for indentation depths up to 1-2 [micro]m (as indicated by the results in the subfigure in Figure 6). In doing so, the experimentally determined indentation load P was fitted, using the least square method, to a [h.sup.3/2] function and the elastic stiffness was then determined from equation (3). The average results from a large number of experiments were
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
It should be noted that the results for the two different sphere diameters are not separated due to the fact that only small, and inconsistent, differences between the two sets of results were found. Clearly, in equation (6), the results for clay determined from a thin film (film thickness 30 [micro]m) are fairly close to the corresponding one from the uniaxial tests while the two sets of results differ substantially for calcium carbonate. However, remembering the uncertainties involved concerning the influence from lower boundary, plasticity, and also friction, any firm conclusions as regards to the appropriateness of microindentation as a tool for material characterization of the materials at issue cannot be drawn without further information from the numerical investigation presented below.
The present numerical investigation is performed in order to determine the influence from boundary effects, plasticity, and also friction on the experimental results discussed above. In doing so, the finite element method is used. The mechanical problem to be investigated then concerns a homogeneous thin film of thickness, t, placed on a rigid substrate and indented in the normal direction by a rigid spherical indenter with radius R at quasi-static conditions (Figure 7). In most calculations, the film is assumed to be perfectly clamped to the substrate but other boundary conditions are also investigated. Friction between the indenter and the material is accounted for in some of the calculations and the resulting governing equations are based on a large strain formulation of the problem. Furthermore, the thin film material is assumed to be isotropic, rate-independent, and either perfectly elastic or classical elastic-plastic. In the latter case, the constitutive specification is given by the Prandtl-Reuss equations with isotropic strain-hardening reading, by aid of the Kronecker identity tensor [[delta].sub.ij],
[FIGURE 7 OMITTED]
[^.[tau].sub.ij] = [E/[(1 + v)]]([[delta].sub.ik][[delta].sub.jl] + [v/[(1 - 2v)]][[delta].sub.ij][[delta].sub.kl] - [[3[[tau]'.sub.ij][[tau]'.sub.kl](E/(1 + v))]/[2[[tau].sub.e.sup.2]([2/3]K + [E/[1 + v]])]])[dot.[epsilon].sub.kl]. (7)
In equation (7), E and v are Young's modulus and Poisson's ratio, respectively; [dot.[epsilon].sub.ij], is the rate of deformation and [^.[tau].sub.ij] is the Jaumann rate of the Kirchhoff stress, [[tau].sub.ij]. The Kirchhoff stress is related to the Cauchy stress, [[sigma].sub.ij], as [[tau].sub.ij], = J[[sigma].sub.ij'] where J is the ratio of volume in current state to volume in previous state. Furthermore, [[tau].sub.e] and [[tau]'.sub.ij] are Mises effective stress and deviatoric stress, respectively. Finally then, in equation (7), K is the instantaneous slope of the uniaxial Kirchhoff stress, [tau], versus the logarithmic accumulated plastic strain, [[epsilon].sub.p]. The constitutive specification in equation (7) is only valid at plastic loading when [[tau].sub.e] = [tau]([[epsilon].sub.p]). At elastic loading, or at elastic unloading, a hypoelastic formulation based on the first part of equation (7) is relied upon. The materials investigated in the numerical study are described in Table 1, with explicit values on elastic properties and initial yield stress being representative of the materials at issue. The strain-hardening behavior was assumed to be either perfectly plastic or linear, as specified in Table 1.
In order to complete the boundary value problem, it remains to be stated that throughout the analysis equilibrium equations, in absence of body forces, have to be satisfied. In the contact region, unilateral constraints, given by the shape of the rigid indenter, are imposed, and in some cases, Coulomb friction, with the friction coefficient denoted [mu], is assumed. Outside the contact area the surface of the film is traction free while at the film/substrate interface--remembering that the film is substantially softer than the substrate--zero displacements are enforced in most of the finite element simulations, as will be discussed in more detail. The indentation load is given by the relation:
P = - [[integral].[[GAMMA].sub.c]][[sigma].sub.2j][N.sub.j]d[[GAMMA].sub.c] (8)
where [[GAMMA].sub.c] is the actual area of contact and [N.sub.j] is the inward unit normal vector to the rigid surface of the indenter.
From a mathematical point of view, the mechanical problem outlined above is very complicated, with both geometrical and material nonlinearities present. As a result, numerical methods have to be relied upon and, more specifically, the commercial finite element program ABAQUS (19) was used for this purpose. The finite element mesh is shown in Figure 8. It consists of 1565 axisymmetric solid elements and 1697 nodes. The displacements were approximated using bilinear shape functions. In case of elastic-plastic deformation, hybrid elements were used, with the hydrostatic pressure attaining a constant value in each element, in order to improve convergence at close to incompressible deformation.
The intention of the present numerical investigation was to investigate different aspects of the experimental procedure described above. In doing so, the metal foundation was approximated as a rigid body, as indicated in Figure 8. In most of the calculations, as already mentioned, clamped boundary conditions were imposed on all nodal points in contact with the lower boundary (the metal foundation) of the mesh, as this is the most favorable situation from a numerical point of view. However, in order to also investigate the effect of sliding and loss of contact along the interface, other boundary conditions were also investigated. In short, even for the extreme case of no friction and no bonding between the film and the foundation, the numerical results for relevant quantities showed essentially total agreement with the corresponding results for a situation with clamped boundary conditions imposed on all nodal points in contact with the metal foundation. Finally, it should also be mentioned that it was shown by Mitre Peterson and Larsson (14) that, in the present circumstances, a contact area between the material and the indenter consisting of approximately 10 elements was required in order to achieve high accuracy of the numerical results. This finding was accounted for in all of the present numerical simulations.
The outcome of the finite element calculations for one of the combinations of film thickness and indenter radius investigated are summarized in Figure 9. It is interesting to note that almost immediately at contact, the explicit value on the indentation load is affected by the presence of the finite thickness of the film. For example, at (h/t) = 0.01, the numerically determined indentation load attains a value close to, or even smaller than, half the value predicted by Hertz theory, equation (3). A ratio (h/t) = 0.01 corresponds to indentation depths 0.3 [micro]m and 0.7 [micro]m, respectively for the 30 and 70 [micro]m-thick films used in the experiments, and such values on the maximum indentation depth are much too small in order to get good accuracy on the experimentally determined modulus of elasticity. As a consequence of this finding, Hertz theory, as presented above, is not sufficient for the present purpose. This is perhaps a somewhat surprising result, but is confirmed by previous and similar investigations. (14) It should be noted in passing, though, that this effect is particularly noticeable at purely elastic deformation and is significantly reduced at predominantly plastic conditions.
A straightforward solution to the problem would be to use the numerical results presented in Figure 9 (and the corresponding ones for other combinations of film thickness and indenter radius) in order to correct the experimentally determined indentation loads for boundary effects. This is a possible scheme, as numerical experiments showed that the results in Figure 9 for purely elastic deformation are very weakly dependent on the explicit values of the elastic material constants (indeed, the curves in Figure 9 are independent of the modulus of elasticity). Consequently, for a fixed value on the ratio (R/t), a relation
[FIGURE 8 OMITTED]
[P.sub.H] = f(h/t)P (9)
can be used in order to correct the experimental results for boundary effects. In equation (9), P is the experimentally determined indentation load, [P.sub.H] is the corresponding indentation load given by Hertz theory, equation (3), and f(h/t) is a correction function given by the elastic numerical results in Figure 9. Based on this procedure, equation (4) can be used in a straightforward manner, as discussed previously, with the indentation load P given by the experiments replaced by [P.sub.H], in order to determine the elastic properties of the materials with boundary effects accounted for. It should be clearly stated that the relation ([P.sub.H]/P) in a general situation also depends on the ratio (R/t) (as mentioned above, in equation 9) [(R/t) is assumed to be fixed]. At very high values on (R/t), the corrections due to boundary effects will be so large that the procedure proposed above can no longer be applied. However, judging from the numerical results in Figure 9 and from the results presented below, the presently used values on (R/t) are acceptable.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
Before presenting the explicit results based on equation (9), it seems appropriate to also discuss the effect on the indentation load from plasticity and friction. With regards to plastic effects, P-h curves pertinent to different values on the initial yield stress and on different strain-hardening behavior, as specified in Table 1, are also presented in Figure 9. Clearly, in the present circumstances, the outcome of the indentation tests are strongly dependent on the initial yield stress, but not so much on the strain-hardening characteristics--and any correction procedure based on equation (9) can only be applied to experimental quantities unaffected by plasticity. As these material characteristics are not known in advance, some caution is definitely recommended when considering which experimental data points to use in order to determine purely elastic material properties. When it comes to frictional effects, the finite element calculations showed that this feature is almost negligible in comparison with the influence from the metal foundation and from plasticity and will not be discussed further in this context.
The correction procedure based on equation (9), and the numerical results in Figure 9, were applied to the experimental results from the indentation tests. In doing so, the experimental data points within the elastic regime, as determined from the numerical investigation, were corrected and fitted to an [h.sup.3/2] function by using the least square method. In this context, it should be immediately mentioned that in order to secure a sufficient amount of data points for the fitting operation, some experimental results influenced by plastic deformation were also included in the correction process. These values were also properly corrected by using, based on the uniaxial tensile tests presented in Figure 5, the numerical results for material 2 in Figure 9. The average results from this procedure were
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
When compared to the uniaxial results in equation (5), the value on the elastic stiffness for calcium carbonate is fairly well determined by the indentation experiments, particularly when remembering that plastic effects definitely influenced the experimental results and uncertainties with regards to the explicit value on Poisson's ratio for the coating materials. When it comes to clay, however, the situation is much worse, as no agreement at all is found between the tensile stiffnesses reported in equations (5) and (10). A possible explanation for this discrepancy can be found when considering the microstructure of the two coating materials, as the clay particles are flat flake-shaped while the calcium carbonate ones have a more isometric (spherical or ellipsoidal) shape (Figure 4). For this reason, it can be expected that clay exhibits a much more anisotropic material behavior and, accordingly, Hertz contact theory, as utilized presently, is not applicable. In addition, the latex binder will contribute to some extent to the elastic stiffness in the thickness direction for materials with flake-shaped particles, cf. e.g., Parpaillon et al., (20) as also indicated by the results for clay in equation (10) (the elastic stiffness of latex is well below 1 GPa, cf. e.g., Pawlak and Keller (6)).
The results described above indicate that microindentation can be a useful tool for determining the elastic properties of paper coating materials. It is obvious, however, that isotropic material behavior, which cannot always be expected, is a necessary condition for the success of the proposed experimental technique. Plastic effects and the influence from a finite film thickness are other features that can complicate the interpretation of the experimental results. An obvious remedy to both these problems is to use films as thick as possible, but it is then necessary to ensure that the composition of the films is homogeneous. It is also important to emphasize the need to secure a larger number of experimental data points at low values of the indentation load, i.e., the elastic regime. This matter was not properly considered during the present investigation but could be handled in the correction procedure as the plastic characteristics of the materials were fairly well known from the uniaxial tensile tests. Such an approach, however, is not possible in a situation where very limited knowledge about the material behavior is at hand prior to indentation.
DETERMINATION OF THE FAILURE STRESS
In case of indentation of brittle materials, it is well known from a number of previous investigations that cracks may appear at (or just outside) the contact boundary, cf. e.g., Cook and Pharr. (21) Most often, the cracks are circumferential as a result of tensile radial stresses. In a truly Hertzian situation, the axisymmetric surface stresses can be expressed in closed form as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
and it is then a fairly straightforward task to determine the failure stress from indentation experiments. There are, however, many features that complicate this procedure and presently this, again, concerns the effect from friction, plasticity, and the lower boundary of the film.
The influence on the radial surface stress from the features mentioned above were also investigated during the finite element calculations, and some limited results for elastic material behavior are shown in Figure 10. It is clear that boundary effects alone will lead to large deviations from the Hertz solution, equation (11). This finding indicates that it is practically impossible to determine the failure stress of the coatings without detailed knowledge of the elastic, and also the plastic, material behavior. If this is the case, the experimental situation at crack initiation can be compared with numerically determined stresses in order to determine the failure stress of the material. In practice, however, such an approach can serve as a multiaxial verification technique of results from other test methods, most likely uniaxial tensile tests, but not as a single alternative for failure stress determination, as it requires full knowledge of the constitutive behavior of the coatings.
Microindentation as a tool for material characterization of mineral-filled latex polymers has been investigated. The most important findings can be summarized as follows:
* Indentation must be performed on single films of coating materials, as otherwise the experimental results will be strongly influenced by the mechanical behavior of the paper sheet.
* The microstructure of the coating materials must be investigated prior to indentation in order to ensure that isotropic material behavior can be expected. Anisotropy will strongly reduce the accuracy of the presently proposed procedure.
* Plastic effects, as well as the influence from the finite thickness of the films, must be accounted for when interpreting the experimental results and this can be achieved by the use of numerical simulations based on the finite element method. In addition, both these effects can be reduced by using films as thick as possible in the experiments, but it is then imperative to ensure that the microstructure of the films are uniform and similar to the microstructure of the thin films.
* The use of indentation, in order to determine the failure stress of the coating materials, is a very involved procedure as it requires full knowledge of the material behavior. In practice, such an approach can only serve as a method for verification of results from other test methods, most likely uniaxial tensile tests.
Table 1 -- Explicit Values of the Material Constants Used in the Numerical Simulations Material No. Explicit Values on the Material Constants 1 Purely elastic material with E = 5 GPa, v = 0.3 2 E = 5 GPa, v = 0.3, [[sigma].sub.y] = 5 MPa, [[sigma].sub.0] = 0 3 E = 5 GPa, v = 0.3, [[sigma].sub.y] = 10 MPa, [[sigma].sub.0] = 0 4 E = 5 GPa, v = 0.3, [[sigma].sub.y] = 10 MPa, [[sigma].sub.0] = 1 MPa 5 E = 5 GPa, v = 0.3, [[sigma].sub.y] = 10 MPa, [[sigma].sub.0] = 10 MPa 6 E = 5 GPa, v = 0.3, [[sigma].sub.y] = 100 MPa, [[sigma].sub.0] = 0 In addition to the numerical calculations pertinent to the materials in the table, a number of finite element simulations were performed with different values on the elastic material constants E and v and on the coefficient of friction [mu], in order to determine the influence from these properties on the correction curves presented in Figure 9. In the table, linear strain-hardening behavior according to the relation [sigma]([E.sub.[rho]]) = [[sigma].sub.y] + [[sigma].sub.0][E.sub.p] is assumed.
Part of this investigation was supported by the Swedish Foundation for Strategic Research, through the Surface Science for Printing Program (S2P2), which is gratefully acknowledged.
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Christophe Barbier, Per-Lennart Larsson,** and Soren Ostlund -- KTH Solid Mechanics*
Nils Hallback ([double dagger]) and Michael Karathanasis -- Swedish Pulp and Paper Research Institute ([dagger])
* Royal Institute of Technology (KTH), SE-100 44, Stockholm, Sweden.
[dagger] P.O. Box 5604, SE-114 86, Stockholm, Sweden.
** Author to whom correspondence should be addressed. Email: email@example.com.
[double dagger] Presently at the Division for Engineering Sciences, Physics and Mathematics, Karlstad University, SE-651 88, Karlstad, Sweden.
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