**3. Experimental**

The materials used include two types. One type is copper for indentation penetration zone measurement. The other one is a medium carbon steel with inclusions for simulated surface damage propagation analysis. A hardened tool carbon steel by heat treatment was used to make the wedge indenter. The indentation configuration is shown in Figure 3. The indenter has a 90o apex angle. The indentation process was conducted under cyclic loading conditions. During indentation, the load and the displacement was recorded by an Xplorer GLX data acquisition unit. These data can be used to plot and show the relation of the indentation load v.s. the nominal indenter penetration depth.

Fig. 3. Indentation set-up for performing simulated surface contact damage tests.

There exists difficulty in measuring the actual damage zone size by direct visual observation. We examined indented copper crystal using scanning electron microscopy (SEM) and measured the damage zone size. The copper polycrystal was etched in warm HCl/SnCl4 solution. Further investigation of the damage zone using electron backscattering diffraction (EBSD) technique to reveal the contact damage zone in single crystal copper was also performed.

### **4. Results and discussion**

150 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

 

The quantities *J\**, d*a*/d*N*, and *a*, can be obtained from indentation experiments. The relationship expressed in Eq. (10) can be plotted in a two dimensional domain, directly giving the value of the specific energy of damage, *γ*, which is the intercept of the straight line. *γ* can be used as a material property related parameter. By examining Eq. (10), as the contact damage propagates, the energy release rate increases, thus the change of the left term *J\*/a* can be leveled by both the increasing of *J\** and the indentation penetration depth, *a*. The variation of the term in the right side of Eq. (10), *D/*[*a*(*da/dN*)], depends on several factors. These are the indentation depth, *a*, the indentation speed, d*a*/d*N* and *D*, the cyclic rate of energy associated with the damage formation. The indentation speed changes with the indentation depth. From energy balance analysis, it is clear that the value of *D* changes with the indentation depth, *a*. Thus, the variation of *D* is well balanced by the change in both *a* and d*a*/d*N*. Thus, on the *J\*/a* vs *D/*[*a*(*da/dN*)] plot, a straight line which is almost parallel to

The materials used include two types. One type is copper for indentation penetration zone measurement. The other one is a medium carbon steel with inclusions for simulated surface damage propagation analysis. A hardened tool carbon steel by heat treatment was used to make the wedge indenter. The indentation configuration is shown in Figure 3. The indenter has a 90o apex angle. The indentation process was conducted under cyclic loading conditions. During indentation, the load and the displacement was recorded by an Xplorer GLX data acquisition unit. These data can be used to plot and show the relation of the

(10)

*J D a da a dN*

 

\*

the *D/*[*a*(*da/dN*)] axis can be obtained.

indentation load v.s. the nominal indenter penetration depth.

Fig. 3. Indentation set-up for performing simulated surface contact damage tests.

**3. Experimental** 

The indentation cyclic load vs time is shown in Figure 4(a). Time-dependent indentation penetration depth was recorded and shown in Figure 4(b). The relation of the indentation load v.s. the indenter penetration depth at a typical cycle is shown in Figure 4(c). From the load-displacement curves, we can calculate the potential energy and the hysteresis energy associated with the contact damage processes as schematically shown in Figure 4(d). The indentation penetration depth, *a*, versus the number of indentation cycles, *N*, for three steels was plotted. The slope of the *a* versus *N* curves was used to calculate d*a*/d*N*, and establish the relationship of indentation speed, d*a*/d*N*, and indentation depth, *a*.

Fig. 4. Calculating energy dissipation terms from indentation test data: (a) cyclic loading profile, (b) time-dependent displacement, (c) load-displacement relationship, (d) illustration showing how to determine the potential energy and hysteresis energy.

Energy Dissipation Criteria for Surface Contact Damage Evaluation 153

The fatigue crack propagation speed versus the energy release rate for the three steels is shown in Figure 5(c). Steel *A* displayed the highest crack growth speed in the entire energy release rate range. In most part of the energy release rate range, for say, *J\** less than 12 kJ/m2, steel *B* and *C* have the crack speed very close to each other. In the energy release rate range of higher than 12 kJ/m2, *B* has higher crack speed than *C*. It can also be seen from Figure 5(c) that the three curves display the similar two-stage crack growth behavior which are corresponding to the stable crack growth stage and the unstable crack growth stage of the specimens from the three steels. A threshold stage was observed only in the pre-crack initiation stage for *A* and *B*. But it extended to the beginning of the stable crack propagation stage for the specimens from *C*. In the stable crack propagation stage, the decreased acceleration in crack speed is an indicative of material damage within the area in front of the

The damage tolerance is evaluated by the specific energy of damage *γ*. The parameters *γ* was calculated using the experimental data generated from fatigue tests including *a*, d*a*/d*N*, *J\**, and *D*. A plot of *J\*/a* versus *D/*[*a*(*da/dN*)] can be generated for each material. Based on the results of the three steels, *A*, *B* and *C*, we generated Figure 6. Three straight lines which are almost parallel to the horizontal axis were obtained for the three steels. The intercepts of the three lines give the values of *γ* for each layer. From the results shown in Figure 6, the value of *γ*, being a material property related parameter, is suitable for characterizing the fatigue

Fig. 6. Plot for determining the specific energy of damage, *γ*, of the three medium carbon steels under different heat treatment conditions: (a) steel A, (b) steel B, (c) steel C.

damage tolerance for evaluating the resistance to fatigue crack growth.

Due to the microstructure change with heat treatment conditions, the specific energy of damage for each of the steels is different. Steel *A* with hardening treatment, has the lowest *γ*, while *C* shows the highest *γ* due to tempering treatment. The specific energy of damage of steel *B*, heat treated at very low air cooling rate, is close to that of *C*. Since *γ* is almost a constant for each material tested, it can be taken as a parameter characteristic of the fatigue

Although the indentation penetration depth is fairly straightforward to be recorded, it is challenge to measure the actual damage zone size. Figure 7(a) is the scanning electron microscopic (SEM) image of the copper polycrystal after etching in warm HCl/SnCl4 solution. It can be seen that the grain boundaries are etched away by the solution. The precision polishing helped to expose the etching pits and islands on the surface of the

crack tip associated with fatigue crack propagation.

damage tolerance.

The potential energy, *P*, was calculated from the loading and unloading curves recorded at intervals of number of cycles as the area above the unloading curve (see Figure 4(d)). On this basis, the relationship between the potential energy and the indentation depth, *a*, can be established. The relationship between *P* and *a* is used to determine the energy release rate, *J\**, using Eq. (8). The hysteresis energy at each indentation cycle *Hn* is determined from the area of the hysteresis loop recorded as schematically shown in Figure 4(d). Based on the value of hysteresis energy and the relationship between *a* versus *N*, the quantity of *D*, the cyclic rate of energy dissipation into contact damage zone evolution is determined using Eq. (9).

Figure 5 shows the fatigue crack growth behavior of three medium carbon steels (named as materials *A*, *B* and *C*) due to the interface debonding of particle inclusions and the pearlite matrix. The carbon content of the three steels is 0.77% in weight. However, the heat treatment conditions are not the same, which affected their fatigue property. Steel *A* was heat treated at the highest cooling rate. *B* has a much lower cooling rate than *A*, while *C* has an even lower cooling rate, but close to that of *B*. Tension-tension fatigue tests with cyclic loading ratio of *R* = 0*:*1 were performed. It is found that the energy release rate and the cyclic energy dissipation rate change constantly for each of the materials during the fatigue crack growth. We also found that the critical value of energy release rate is very difficult to determine as shown in Figure 5(a), the energy release rate, *J\**, versus the crack length for the three steels. The increase of the crack length, *a*, causes the increase of the values of *J\** for the three steels. Therefore, the energy release rate can not be considered as a materials parameter for comparing the fatigue damage tolerance of different materials because a unified value for each material can not be found.

Fig. 5. The fatigue crack propagation data of three medium carbon steels: (a) the energy release rate, *J\**, versus the crack length, *a*, (b) the cyclic rate of energy dissipation, *D*, versus the crack length, *a*, (c) crack speed versus the energy release rate.

The irreversible energy dissipation during fatigue damage of the three steels was also calculated. Based on the measured hysteresis energy for both notched and unnotched specimens and the relationship between crack length *a* versus fatigue cycle *N*, the quantity of *D*, the cyclic rate of energy dissipation into damage zone evolution was determined. The relationships of *D* and the crack length, *a*, for the steels, are shown in Figure 5(b). Material *A*  displayed much higher value of the cyclic rate of energy dissipation into the active zone evolution. The other two steels, *B* and *C* demonstrated very similar behavior. For all of the three steels, it is evident that with the increase in crack length, the values of *D* increase.

The potential energy, *P*, was calculated from the loading and unloading curves recorded at intervals of number of cycles as the area above the unloading curve (see Figure 4(d)). On this basis, the relationship between the potential energy and the indentation depth, *a*, can be established. The relationship between *P* and *a* is used to determine the energy release rate, *J\**, using Eq. (8). The hysteresis energy at each indentation cycle *Hn* is determined from the area of the hysteresis loop recorded as schematically shown in Figure 4(d). Based on the value of hysteresis energy and the relationship between *a* versus *N*, the quantity of *D*, the cyclic rate of energy dissipation into contact damage zone evolution is determined using Eq. (9).

Figure 5 shows the fatigue crack growth behavior of three medium carbon steels (named as materials *A*, *B* and *C*) due to the interface debonding of particle inclusions and the pearlite matrix. The carbon content of the three steels is 0.77% in weight. However, the heat treatment conditions are not the same, which affected their fatigue property. Steel *A* was heat treated at the highest cooling rate. *B* has a much lower cooling rate than *A*, while *C* has an even lower cooling rate, but close to that of *B*. Tension-tension fatigue tests with cyclic loading ratio of *R* = 0*:*1 were performed. It is found that the energy release rate and the cyclic energy dissipation rate change constantly for each of the materials during the fatigue crack growth. We also found that the critical value of energy release rate is very difficult to determine as shown in Figure 5(a), the energy release rate, *J\**, versus the crack length for the three steels. The increase of the crack length, *a*, causes the increase of the values of *J\** for the three steels. Therefore, the energy release rate can not be considered as a materials parameter for comparing the fatigue damage tolerance of different materials because a

Fig. 5. The fatigue crack propagation data of three medium carbon steels: (a) the energy release rate, *J\**, versus the crack length, *a*, (b) the cyclic rate of energy dissipation, *D*, versus

The irreversible energy dissipation during fatigue damage of the three steels was also calculated. Based on the measured hysteresis energy for both notched and unnotched specimens and the relationship between crack length *a* versus fatigue cycle *N*, the quantity of *D*, the cyclic rate of energy dissipation into damage zone evolution was determined. The relationships of *D* and the crack length, *a*, for the steels, are shown in Figure 5(b). Material *A*  displayed much higher value of the cyclic rate of energy dissipation into the active zone evolution. The other two steels, *B* and *C* demonstrated very similar behavior. For all of the three steels, it is evident that with the increase in crack length, the values of *D* increase.

the crack length, *a*, (c) crack speed versus the energy release rate.

unified value for each material can not be found.

The fatigue crack propagation speed versus the energy release rate for the three steels is shown in Figure 5(c). Steel *A* displayed the highest crack growth speed in the entire energy release rate range. In most part of the energy release rate range, for say, *J\** less than 12 kJ/m2, steel *B* and *C* have the crack speed very close to each other. In the energy release rate range of higher than 12 kJ/m2, *B* has higher crack speed than *C*. It can also be seen from Figure 5(c) that the three curves display the similar two-stage crack growth behavior which are corresponding to the stable crack growth stage and the unstable crack growth stage of the specimens from the three steels. A threshold stage was observed only in the pre-crack initiation stage for *A* and *B*. But it extended to the beginning of the stable crack propagation stage for the specimens from *C*. In the stable crack propagation stage, the decreased acceleration in crack speed is an indicative of material damage within the area in front of the crack tip associated with fatigue crack propagation.

The damage tolerance is evaluated by the specific energy of damage *γ*. The parameters *γ* was calculated using the experimental data generated from fatigue tests including *a*, d*a*/d*N*, *J\**, and *D*. A plot of *J\*/a* versus *D/*[*a*(*da/dN*)] can be generated for each material. Based on the results of the three steels, *A*, *B* and *C*, we generated Figure 6. Three straight lines which are almost parallel to the horizontal axis were obtained for the three steels. The intercepts of the three lines give the values of *γ* for each layer. From the results shown in Figure 6, the value of *γ*, being a material property related parameter, is suitable for characterizing the fatigue damage tolerance.

Fig. 6. Plot for determining the specific energy of damage, *γ*, of the three medium carbon steels under different heat treatment conditions: (a) steel A, (b) steel B, (c) steel C.

Due to the microstructure change with heat treatment conditions, the specific energy of damage for each of the steels is different. Steel *A* with hardening treatment, has the lowest *γ*, while *C* shows the highest *γ* due to tempering treatment. The specific energy of damage of steel *B*, heat treated at very low air cooling rate, is close to that of *C*. Since *γ* is almost a constant for each material tested, it can be taken as a parameter characteristic of the fatigue damage tolerance for evaluating the resistance to fatigue crack growth.

Although the indentation penetration depth is fairly straightforward to be recorded, it is challenge to measure the actual damage zone size. Figure 7(a) is the scanning electron microscopic (SEM) image of the copper polycrystal after etching in warm HCl/SnCl4 solution. It can be seen that the grain boundaries are etched away by the solution. The precision polishing helped to expose the etching pits and islands on the surface of the

Energy Dissipation Criteria for Surface Contact Damage Evaluation 155

to the stress states in regions underneath the indented zone are obtained. Instrumental indentation performed on copper materials with different grain sizes reveals both the indentation zone and damage zone. The reason for choosing copper is the high ductility of copper which allows deformation develops in a stable way during the indentation

Based on the experimental studies of fatigue crack growth on three steels, the criterion for evaluating the extent of damage is identified. Although it is difficult to find a unified stress or strain based damage criterion to characterize the damage evolution, energy dissipation analysis provides a more accurate way to describe the deformation behavior of the materials. Under wedge indentation, the analysis shows advantage because the stress field has the singularity which limits the applicability of the strength criterion. The loaddisplacement relations with elastic-plastic responses of the materials associated with the indentation processes were obtained. The hysteresis energy was also determined. Lattice rotation measurement using electron backscatter diffraction (EBSD) technique in the region ahead of the indenter tip is an effective way to measure the dimension of the contact damage zone (CDZ) and the results can be used to define the length scales during contact deformation. A unified criterion using the hysteresis energy normalized by the length scales has been established. The above mentioned indentation tests in this work caused deformation of significant amount of materials. For further studies, comparison of deep

[1] Kawaguchi, T., & Pearson, R.A. (2004). The moisture effect on the fatigue crack growth of

[2] Kawaguchi, T. & Pearson, R.A. (2004). The moisture effect on the fatigue crack growth of

[3] Shi, G., Zhang, M.Q., Rong, M.Z., Wetzel, B. & Friedrich, K. (2003). Friction and wear of low nanometer Si3N4 filled epoxy composites, *Wear*. Vol.(254): 784-796 [4] Zhang, R. & Shi, Z. (2008). Bi-interfacial debonding of coated fiber reinforced composites under fatigue load, *International Journal of Fatigue.* Vol.(30): 1074-1079 [5] Kushch, V.I., Shmegera, S.V. & Mishnaevsky, L. (2008). Meso cell model of fiber

[6] Cavallini, M., Di Bartolomeo, O.D. & Iacoviello, F. (2008). Fatigue crack propagation

[7] Chan, K.S., Lee, Y.D., Nicolella, D.P., Furman, B.R., Wellinghoff, S. & Rawls, R. (2007).

and micromechanics, *Engineering Fracture Mechanics*. Vol.(74):1857-1871 [8] Sharratt, B.M., Wang, L.C. & Dauskardt, R.H. (2007). Anomalous debonding behavior of

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damaging micromechanisms in ductile cast irons, *Engineering Fracture Mechanics*.

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indentation and nanoindentation should be performed.

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Vol.(75): 694-704

processes.

**6. References** 

specimen. These features come from the selectively dissolving of materials located near the ends of the dislocation lines. However, the indented damage zone is still unclear.

Further investigation of the damage zone using electron backscattering diffraction (EBSD) technique reveals different features within the contact damage zone. For example, the band contract map, Figure 7(b), provides the features of subgrain formation and recrystallization of the single crystal grain under wedge indentation after annealing. Since the intensity of the backscatter electrons changes from grain to grain, the grain boundary can be revealed by the band contrast change. Thus, it is possible to identify the microstructure in the area close to the indentation tip. By this method, the subgrain formation due to severe contact damage and plastic deformation can be revealed. The average size of the subgrains shown in Figure 7(b) is about 10 to 15 µm. It is also found there is an elliptical region in front of the indentation tip, which corresponds to the strain hardened elastic-plastic zone. Deeper into the indentation region, it is the fully plastic deformation zone, as shown by the in-plane lattice rotation map in Figure 7(c). Such EBSD results will provide us the insight into how to determine the size of the damage zone. For example the conservative measurement will give us the size of the damage zone the same as the indenter penetration zone (IPZ) as shown by the elliptical region in Figure 7(b). A more accurate measurement should account for the extended plastic region as shown in Figure 7(c). The distance from point *A* to point *C* or *E*  instead of just from point *O* to *A* should be considered as the damage zone size, which is about 5 times larger than the indenter penetration zone (IPZ). This EBSD measurement results were used to correct the damage tolerance calculation by adding the contact damage zone size to the indenter penetration depth or crack length, *a*. Consequently, the indenter penetration speed d*a*/d*N* was modified as the damage zone expansion speed.

Fig. 7. Measuring the size of indentation contact damage zone via electron microscopy: (a) scanning electron microscopic measurement, (b) band contrast map of the indentation penetration zone obtained by electron backscatter diffraction (EBSD) measurement, (c) in plane lattice rotation map generated by electron backscatter diffraction (EBSD) measurement.

### **5. Conclusions**

The energy dissipation approach is applicable for analyzing surface contact damages in various materials, including composite materials. The contact of different bodies can be modeled as indentation. Analysis of indentation and modeling of the deformation states of indented materials at different scales are performed. The stress distributions within indentation zones are described by fracture mechanics, and single crystal plasticity solutions to the stress states in regions underneath the indented zone are obtained. Instrumental indentation performed on copper materials with different grain sizes reveals both the indentation zone and damage zone. The reason for choosing copper is the high ductility of copper which allows deformation develops in a stable way during the indentation processes.

Based on the experimental studies of fatigue crack growth on three steels, the criterion for evaluating the extent of damage is identified. Although it is difficult to find a unified stress or strain based damage criterion to characterize the damage evolution, energy dissipation analysis provides a more accurate way to describe the deformation behavior of the materials. Under wedge indentation, the analysis shows advantage because the stress field has the singularity which limits the applicability of the strength criterion. The loaddisplacement relations with elastic-plastic responses of the materials associated with the indentation processes were obtained. The hysteresis energy was also determined. Lattice rotation measurement using electron backscatter diffraction (EBSD) technique in the region ahead of the indenter tip is an effective way to measure the dimension of the contact damage zone (CDZ) and the results can be used to define the length scales during contact deformation. A unified criterion using the hysteresis energy normalized by the length scales has been established. The above mentioned indentation tests in this work caused deformation of significant amount of materials. For further studies, comparison of deep indentation and nanoindentation should be performed.
