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## Analytical Methods

Introduction

This chapter explores the analytical techniques which have been used to provide a quantitative measure of the adhesion in the epoxy coating/Si substrate system, using the probe test technique. The interfacial fracture energy, *G** _{c}* (or critical strain energy release rate), provides an intuitively meaningful measure of the energy required to debond a unit area of an interfacial crack in an adhesive bond and hence characterizes the resistance of the adhesive bond to debond by fracture. Using classical analytical solutions, the probe test geometry, as described in Chapter 2, has been modeled to calculate the interfacial fracture energy for the epoxy coating/Si substrate system using two approaches: shaft – loaded blister approximation and curvature method.

loaded blister approximation

The probe test experiments were performed on a Nikon UM-2 Measurescope microscope with a digital measuring stage, as described in Chapter 3. A typical debond created by lifting the edge of a coating is shown in Figure 4-1. The debond was created as the conical probe tip was forced under the coating. The radii of the debond along and perpendicular to the probe direction were recorded for a particular value of probe penetration distance. It was shown in Chapter 3 that the crack front of the debond was approximately semi-circular in shape. Closed form solutions for the interfacial fracture energy can be obtained [1] for a circular shaft – loaded blister geometry, shown in Figure 4-2. Thus, in order to develop an analytical model for the debond in the probe test geometry, the analytical expressions from the circular shaft – loaded blister geometry were used as an approximation for the probe test geometry. It should be noted that the geometry of the debond in the probe test geometry is similar to a half – blister with a free edge as compared to the approximation used for a circular blister geometry. The geometry of a shaft – loaded circular blister can be analyzed as a circular plate loaded at the center . Since the blister is bonded ahead of the crack tip, a clamped boundary condition can be assumed at the edges of the circular plate. Depending on the maximum deflection (*w*_{0 }) and radius (*a *) of the plate relative to its thickness (*h *), the following classifications are used in literature: For a linearly elastic thin plate, the analytical solutions for small and large deflections of a clamped circular plate have been presented by Timoshenko . The following analysis is presented using a thin plate formulation for small and large deflections of the plate. Also, a correction for thick plates is presented.

Theory

Thin plate – small deflections

Assuming linear elasticity and small deflections , the deflection profile

### Results and discussion

As mentioned earlier in Section 4.2, the debond in the probe test was semicircular in shape with one edge free and the other edge clamped, however the closedform solutions presented in Section using different plate theory formulations were obtained from a shaft-loaded circular blister geometry with clamped edges. Also, the deflection profile or shape of a circular blister is different from the deflection profile of the debonded coating in the probe test. The gradient of the blister surface at the center of the blister (*r *= 0) is zero; however the debonded coating surface has a finite gradient at the corresponding point (*r *=0). Thus, this approximation of the probe test geometry by a circular blister geometry is somewhat crude and may possibly introduce an error in the results presentated in this section. For the probe test geometry shown in Figure 4-3, δ is the probe penetration distance, *a* is the debond radius along the probe direction, *h* is the thickness of the coating and *w*_{0} is the maximum vertical separation distance of the coating from the substrate at the point of intrusion. In the probe test experiments, the probe tip was inclined at an angle of 25.0° ± 1.0. From the geometry, including the half cone angle of the probe (3°); *w*_{0} can be related to δ as: Using the probe test experiments, measurements were obtained for 5 different debonds created on the specimen; for each debond the measurements were collected for different values of probe penetration distances. Figure 4-4 shows the debond radius (*a *) recorded as a function of the probe penetration distance (δ) for the different debonds created on the coating/Si specimen using a probe angle of 25.0°. Using Equation (4.18), the maximum deflection (*w*_{0 }) of the coating was calculated from the different probe penetration distance (δ) values. Figure 4-5 shows the *a *normalized debond radius as a function of normalized coating deflection . It was observed that all the data points lie in the region > 0.5and <10. This showed that the deformations of the coatings lie in the domain of large deflection and thick plate theory. Using Equation (4.9) for thin plate – small deflection theory, the interfacial fracture energy, *G** _{c}* was calculated for different debond sizes. Using Equation (4.14) for thin plate – large deflection theory, the interfacial fracture energy,

*G*

*was calculated for different debond sizes. Figure 4-7 shows the*

_{c}*G*

*values as a function of debond radius for five different debonds. For thick plate – large deflection theory, the*

_{c}*w*

_{0}values were divided by the correction factor C as given in Equation (4.16). Then, using Equation (4.17) for large deflection theory, the interfacial fracture energy,

*G*

*was calculated for different debond sizes. Figure 4-8 shows the*

_{c}*G*

*as a function of debond radius for five different debonds. It was observed from Figure 4-6, Figure 4-7 and Figure 4-8 that the*

_{c}*G*

*values vary considerably for the five different debonds that were created on the same specimen. This variation in*

_{c}*G*

*values could be due to variability of the experimental test method like change in probe angle due to flexing of the probe shaft, difference in the probe incremental displacement, etc. Also, since the debonds were created at different locations in the specimen, small variations in the coating mechanical properties (*

_{c}*E*,ν) and coating thickness (

*h*) at the different debond locations could lead to variation in the calculated

*G*

*values. As it can be seen from Equations (4.2), (4.9), (4.14) and (4.17),*

_{c}*G*

*is proportional to*

_{c}*h*

^{3}, thus a small variation in coating thickness (

*h*) would introduce a considerable error in the predicted

*G*

*values. However, this local variation of the coating thickness (*

_{c }*h*) was not measured experimentally, instead an average thickness (

*h*) value of 70 μm was used for

*G*

*calculations for the five different debonds. Also, the values used for the mechanical properties (*

_{c}*E*,ν) of the coating, as presented in Chapter 3, were obtained using testing of bulk model epoxy adhesive specimen. The in-situ (i.e. when applied to substrate) coating mechanical properties can be affected by the degree of cure, aging effects and environmental conditions (e.g. temperature and humidity). Local changes in these coating mechanical properties (

*E*,ν) at different debond locations were not taken into account in the present work and could result in error in the predicted

*G*

*values. Finally, the*

_{c }*G*

*values for the five different debonds were averaged, for each of the plate theory formulation. Figure 4-9 shows a comparison of the*

_{c}*G*

*values calculated using the different plate theory formulations. In thin plate theory, based on the Love-Kirchhoff approximation, the transverse normal and shear strain in the plate are neglected . Thus for thin plate theory, this results in an over-estimation of*

_{c}*G*

*values while using displacement-based equations for*

_{c}*G*

*. Hence, as shown in Figure 4-9, the*

_{c }*G*

*values predicted using thin plate theories (small and large deflections), (Equations (4.9) and (4.14)), were higher compared with the*

_{c}*G*

*values predicted using thick plate theory (Equation (4.17)). Also, the transverse strains in the coating are higher for smaller values of debond radii (*

_{c}*a*) i.e. smaller debonds. Hence, the resulting error in the

*G*

*values predicted using thin plate theories is much higher for smaller debond sizes since the higher transverse strains are not taken into account. Thus, as shown in Figure 4-6, Figure 4-7 and Figure 4-9, the*

_{c}*G*

*values predicted using thin plate theories (small and large deflections), were higher for smaller debond radii (*

_{c}*a*). Lastly, small deflection theory neglects the stretching deformation of the coating. As the debond grows in size, the in-plane stretching deformation of the coating increases, thus the

*G*

*values are under-predicted for larger debond sizes. Hence, as shown in Figure 4-9, the*

_{c}*G*

*values predicted using thin plate – small deflection formation decreased with increasing debond radii (*

_{c}*a*). The large deflection theory takes into account the in-plane stretching deformation and hence for higher debond sizes predicts higher

*G*

*values compared with small deflection theory. Thus, as shown in Figure 4-9, the*

_{c}*G*

*values predicted using the thin plate – large deflection formulation (Equation (4.14)) and the thick plate – large deflection formulation ((4.17)), gradually increase for increasing debond sizes. To summarize, the thick plate – large deflection formulation takes into account both the stretching deformation (due to large deflection) and transverse strains (due to thick plate) and thus predict lower*

_{c}*G*

*values (as shown in Figure 4-8 and Figure 4-9) compared with thin plate theories that show a gradual increase with increasing debond size. It was mentioned earlier that the correction factor*

_{c}*C*given by Equation (4.16) was based on an empirical approximation without any theoretical foundation. Also, the analytical solution used in Equation (4.17) was obtained using a shaft – loaded circular blister approximation. However, as discussed in Chapter 6, compared with thin plate formulations, the

*G*

*values predicted by the thick plate – large deflection formulation were the closest to*

_{c}*G*

*values obtained using finite element simulations (contact interaction analysis) in Chapter 5. Also, as shown earlier the deformation of the coating was in the domain of large deflection and thick plate theory. Therefore, it can be concluded that the*

_{c}*G*

*values predicted in Figure 4-8 and Figure 4-9 using thick plate – large deflection formulation were the best estimate for the interfacial fracture energy of the epoxy coating (70 μm)/Si system used as the sample system in the present work. The*

_{c}*G*

*values increase with increasing debond sizes, which would indicate that the energy required to drive the crack increases as the debond grows in size and the crack front moves further away from the free edge of the coating.*

_{c}#### Curvature Method

The curvature method provides an alternate method of estimating the interfacial fracture energy of an adhesive system by using the local curvature at the crack tip. This approach was first developed by Obreimoff using elementary beam theory formulation and used to estimate the cohesive strength of freshly split mica foils. Goussev has extended this approach, using a linear plate bending formulation, to calculate the interfacial fracture energy for a blister test geometry and a double cantilever beam geometry ]. The pressurized blister test experiments were conducted by Goussev using two commercial polymer films bonded to polymethylmethacrylate (PMMA) and polytetrafluoroethylene (PTFE) substrates and the blister profile was measured experimentally using a scanning capacitance microscope. Based on the approach developed in [5-7], the following analysis is presented.

1. Introduction

1.1 Overview

1.2 Research Objective

1.3 Outline

1.4 References

2. Literature Review

2.1 Fracture Mechanics

2.2 Fracture-based measurement techniques for thin – film adhesion

2.3 Effect of residual stress on coating delamination

2.4 Summary

2.5 Figures

2.6 References

3. Experimental Procedure

3.1 Introduction

3.2 Specimen Preparation

3.3 Probe Test

3.4 Summary

3.5 Figures

3.6 References

4. Analytical Methods

4.1 Introduction

4.2 Shaft – loaded blister approximation

4.3 Curvature Method

4.4 Summary

4.5 Figures

4.6 References

5. Numerical Methods

5.1 Introduction

5.2 Displacement field-based finite element model

5.3 Contact interaction analysis

5.4 Summary

5.5 Tables

5.6 Figures

5.7 References

6. Summary

6.1 Overview

6.2 Comparison with Xu’s work

6.3 Comparison with Mount’s work

6.4 Summary

6.5 Tables

6.6 Figures

6.7 References

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QUANTITATIVE EVALUATION OF THIN FILM ADHESION USING THE PROBE TEST