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3. Exercise 1 You have developed a new copper alloy that has titanium diboride (TiB2) as second phase particles. The total volume fraction of TIB2 is 3 %. The particle size is uniform throughout the matrix of copper. You made five samples by varying particle size as follows: 0.05 microns, 0.1 microns, 1 microns, 10 microns, and 100 microns . (a) Estimate the hardness of each material. ( b) Estimate the wear coefficient of each sample, assuming that they wear by delamination . State your assumptions clearly. Question

4. Solution- To estimate the hardness and wear coefficient of each material, we need to understand the principles governing these properties in composite materials. We can utilize theoretical models and empirical relationships to estimate the hardness and wear coefficient based on the volume fraction and particle size of the second phase particles (TiB2) in the copper matrix. Part (a): Estimating the Hardness The hardness of a composite material can be estimated using the rule of mixtures or other empirical relationships that take into account the volume fraction and particle size of the reinforcement phase. In this case, we have 3% volume fraction of TiB2 particles in a copper matrix . For metal matrix composites, an empirical relationship to estimate hardness is given by :

5. where: Hc ​ is the hardness of the composite . Hm is the hardness of the matrix (pure copper). Vf is the volume fraction of the reinforcement. d is the particle size . k, α, and β are empirical constants. For simplicity, we assume typical values for copper and TiB2: Hardness of copper ( H_m ): ~100 HV (Vickers hardness ). Empirical constants: k=500k , α=0.5, β=1 ( values are assumed for estimation; actual values can vary based on material properties and experimental data).

6. Given the particle sizes: 0.05 microns 0.1 microns 1 micron 10 microns 100 microns Let's calculate the hardness for each particle size: For d=0.05 microns : Hc ​=100+500⋅0.173⋅20 Hc ​=100+500⋅ 3.46 Hc​= 100+1730 Hc = 1830

7.   For d=0.1microns: Hc ​=100+500⋅0.173⋅ 10 Hc ​=100+500⋅ 1.73 Hc ​= 100+865 Hc ​= 965 For d=1 micron:

8. Hc ​=100+500⋅0.173⋅ 1 Hc ​=100+500⋅ 0.0173 Hc ​= 100+8.65 Hc ​= 108.65 For d=100 microns: Hc ​=100+500⋅0.173⋅ 0.01 Hc ​=100+500⋅ 0.00173 Hc ​= 100+0.865 Hc ​=100.865

9. Summary of Hardness Estimates : 0.05 microns: 1830 HV 0.1 microns: 965 HV 1 micron: 186.5 HV 10 microns: 108.65 HV 100 microns: 100.865 HV Part (b): Estimating the Wear Coefficient Wear coefficient can be estimated using the Archard's wear law, which relates the volume of material worn away to the hardness and other factors. The wear coefficient K is inversely proportional to the hardness : K∝1/H​

10. Assuming the wear coefficient for pure copper ( Km​ ) is known and using the relative hardness values, we can estimate the wear coefficient for the composites. If we assume Km​ for pure copper is 1 (in arbitrary units ): For d= 0.05 microns: For d= 0.1 microns:

11. . For d= 1 micron: For d= 1 micron: For d= 100 micron:

12. Summary of Wear Coefficient Estimates : 0.05 microns: 0.0546 0.1 microns: 0.1036 1 micron: 0.536 10 microns: 0.92 100 microns: 0.992 The hardness of the copper matrix is constant and unaffected by the presence of TiB2 particles.The empirical constants ( k, α, β) are assumed based on typical values for metal matrix composites.The relationship between wear coefficient and hardness is assumed to be inversely proportional, following Archard's wear law.The hardness and wear behavior are considered uniform and isotropic in the material.The particles are uniformly distributed without agglomeration or clustering.

13. Exercise 2- A cam/follower system is designed to control the opening of a valve. To reduce the wear of a cam made of heat-treated 1045 steel, it was coated with a 5 micron thick TiC layer. The follower, which is made of 52100 steel and has a radius of 0.5 inches, slides on the cam. The load applied by the follower on the cam can vary from a low load to extremely high load. Describe how the cam coated with TiC is going to wear as a function of the applied load. Estimate the maximum load that can be applied by the follower without damaging the coating.

14. Solution- Wear of Cam Coated with TiC as a Function of Load Low Load : Adhesive wear will be minimal due to the high hardness of TiC (~3200 HV). TiC provides excellent resistance to wear . Moderate Load : Increased abrasive wear. Possible microfracturing of the TiC coating due to high localized stresses. Risk of coating delamination if microcracks propagate .

15. High Load : Significant risk of fatigue wear and spalling . High stresses may cause underlying 1045 steel to deform, leading to failure of the TiC coating . Estimating Maximum Load Without Damaging the Coating Using Hertzian contact stress theory, assuming a 52100 steel follower (radius = 0.5 inches, modulus ~210 GPa ): Effective Modulus (E ):*

16. . This estimation ensures the applied load does not exceed the fracture toughness of the TiC coating and prevents damage.