**3.4. Computational simulations**

To better understand how the structural multilayering and grading influences the contact loadbearing behavior of the MHSed materials, an elastic–perfectly plastic finite element analysis (FEA) computational model was developed. Here the three-dimensional indenter geometry represented as two dimensional, axisymmetric and rigid has been simulated to fit the experimental nanoindentation loading-depth data for each layer. The extensive study showed that incorporating the post yield strain hardening (linear isotropic, linear kinematic and Ramberg– Osgood isotropic hardening) into the models had a minimal effect on improving the prediction of the simulated data and the estimated yield strength [10]. Consequently, we assume zero hardening for plastic behavior and that the material deformation is elastic–perfectly plastic. This is a simple and effective approach to describe the mechanical behavior of the material.

In these simulations, the FEA-predicted unloading slope was selected to match the experimentally calculated average *E*O–P for each layer (**Figure 3b**). Various material yield strengths (*σY*) were chosen to determine the optimal value at which the FEA-predicted load and unload–depth curve best fits the experimental load and unload–depth curve. The averaged experimental load–depth curves (solid line in **Figure 5**) were selected from the experimental nanoindentation load–depth curve dataset, whose unloading slope represents the average indentation modulus for each layer presented in **Figure 3b**. In the initial simulations, the yield strength of each layer was based on results from our earlier work using micro-compression testing [20]. Using such yield strength, the simulated load–depth curve (dash line in **Figure 5**) was obtained and compared to the averaged experimental curve. By adjusting the yield strength value and iteratively repeating the simulations until the simulated and experimental curves correlated (**Figure 5**), the yield strength for each layer was determined. **Figure 5** shows the FEA simulations best fit the averaged experimental load–depth curve for each component layer. All the component layers exhibit mechanical hysteresis and energy dissipation (calculated as the area of the average experimental load–depth curves), which increased with the distance from the outer layer. The good agreement between the experimental data and the computational simulations gives the material yield strength *σY* values of 1.76, 1.38 and 1.15 GPa for the A/NC

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In our FEA simulations, the effects of the structural multilayering and grading on the largelength-scale mechanical indentation of the MHSed Ti were explored by constructing two axisymmetric two-dimensional FEA models. The first "discrete" model consisted of a MHS of three

**Figure 5.** Experimental and FEA simulated average nanoindentation load–depth curves for the A/NC, NG layers and the

layer, the NG layer and the UFG core, respectively.

UFG core.

**Figure 4.** SEM and OM images of indentation testing results of the MHSed Ti and monolithic NG Ti. (a and b) Top surface SEM views of the MHSed Ti and monolithic NG Ti, respectively. (c and d) Cross-sectional view of the MHSed Ti and monolithic NG Ti, respectively. (e and f) OM cross-sectional views of the plastic deformed area in the MHSed Ti and monolithic NG Ti, respectively. (g and h) HR OM images of the boundary of elastic–plastic strain zone for the MHSed Ti and monolithic NG Ti, respectively.

In these simulations, the FEA-predicted unloading slope was selected to match the experimentally calculated average *E*O–P for each layer (**Figure 3b**). Various material yield strengths (*σY*) were chosen to determine the optimal value at which the FEA-predicted load and unload–depth curve best fits the experimental load and unload–depth curve. The averaged experimental load–depth curves (solid line in **Figure 5**) were selected from the experimental nanoindentation load–depth curve dataset, whose unloading slope represents the average indentation modulus for each layer presented in **Figure 3b**. In the initial simulations, the yield strength of each layer was based on results from our earlier work using micro-compression testing [20]. Using such yield strength, the simulated load–depth curve (dash line in **Figure 5**) was obtained and compared to the averaged experimental curve. By adjusting the yield strength value and iteratively repeating the simulations until the simulated and experimental curves correlated (**Figure 5**), the yield strength for each layer was determined. **Figure 5** shows the FEA simulations best fit the averaged experimental load–depth curve for each component layer. All the component layers exhibit mechanical hysteresis and energy dissipation (calculated as the area of the average experimental load–depth curves), which increased with the distance from the outer layer. The good agreement between the experimental data and the computational simulations gives the material yield strength *σY* values of 1.76, 1.38 and 1.15 GPa for the A/NC layer, the NG layer and the UFG core, respectively.

In our FEA simulations, the effects of the structural multilayering and grading on the largelength-scale mechanical indentation of the MHSed Ti were explored by constructing two axisymmetric two-dimensional FEA models. The first "discrete" model consisted of a MHS of three

100µm

h

**Figure 4.** SEM and OM images of indentation testing results of the MHSed Ti and monolithic NG Ti. (a and b) Top surface SEM views of the MHSed Ti and monolithic NG Ti, respectively. (c and d) Cross-sectional view of the MHSed Ti and monolithic NG Ti, respectively. (e and f) OM cross-sectional views of the plastic deformed area in the MHSed Ti and monolithic NG Ti, respectively. (g and h) HR OM images of the boundary of elastic–plastic strain zone for the MHSed

f **Edge cracks**

**Shear bands**

**Shear bands**

100µm

50µm 50µm

g h

e

86 Contact and Fracture Mechanics

g

**Elastic–plastic strain zone**

Ti and monolithic NG Ti, respectively.

**Figure 5.** Experimental and FEA simulated average nanoindentation load–depth curves for the A/NC, NG layers and the UFG core.

layers with thicknesses matching their experimentally measured values (**Figure 6a**, left). In the discrete model, each layer was assumed to possess isotropic, elastic–perfectly plastic constitutive behavior with *E* and *σY* taken as those calculated from FEA simulations of the averaged loadingdepth data in **Figure 5**. The second "gradient" model is also composed of the three component layers with thickness corresponding to their experimental values and assumed isotropic elastic– perfectly plastic material property, but incorporates linear gradations (**Table 1**) in *E* (scaled by the measured *E*O–P gradation) and *σY* (scaled by the measured *H*O–P gradation) within the material layers (**Figure 6a**, right). **Figure 6b** shows the FEA simulation results of these two models compared with three simulations of the single homogeneous component layer of the A/NC, NG layers and the UFG core. The predications of these two multilayered models show similar load–

**Figure 7.** The simulated strain and stress contours for 1000 N maximum load indentation. (a) Monolithic NG, (b) discrete model, and (c) gradient model. *Sii* (*i* = 1–3) represents the normal stress along *i* axis, and *S*12 is the shear stress that is in

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500µm

a a

b

c

All NG

**3**

**2**

**1**

Gradient

the plane perpendicular to the 1 axis and acts along the 2 axis.

Discrete

depth behavior and both fell in between the simulation of the A/NC and the NG layers.

Based on the FEA simulations, the mechanical behavior of the MHS material was explored by deducing each load–depth performance to an effective O–P modulus, effective microhardness and energy dissipation. The effective modulus and hardness predicted for the two models

**Figure 6.** (a) FEA simulated models of discrete (left) and gradient (right) multilayered structure, the corresponding elastic modulus and yield strength distributions are presented in the center. (b) FEA simulated microindentation load–depth curves for different component layer and two models (discrete and gradient), (c) simulated effective microhardness and experimentally measured values, (d) simulated effective indentation modulus, and (e) simulated effective energy dissipation.

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a

88 Contact and Fracture Mechanics

b c

d e

**Figure 6.** (a) FEA simulated models of discrete (left) and gradient (right) multilayered structure, the corresponding elastic modulus and yield strength distributions are presented in the center. (b) FEA simulated microindentation load–depth curves for different component layer and two models (discrete and gradient), (c) simulated effective microhardness and experimentally measured values, (d) simulated effective indentation modulus, and (e) simulated effective energy dissipation.

**Figure 7.** The simulated strain and stress contours for 1000 N maximum load indentation. (a) Monolithic NG, (b) discrete model, and (c) gradient model. *Sii* (*i* = 1–3) represents the normal stress along *i* axis, and *S*12 is the shear stress that is in the plane perpendicular to the 1 axis and acts along the 2 axis.

layers with thicknesses matching their experimentally measured values (**Figure 6a**, left). In the discrete model, each layer was assumed to possess isotropic, elastic–perfectly plastic constitutive behavior with *E* and *σY* taken as those calculated from FEA simulations of the averaged loadingdepth data in **Figure 5**. The second "gradient" model is also composed of the three component layers with thickness corresponding to their experimental values and assumed isotropic elastic– perfectly plastic material property, but incorporates linear gradations (**Table 1**) in *E* (scaled by the measured *E*O–P gradation) and *σY* (scaled by the measured *H*O–P gradation) within the material layers (**Figure 6a**, right). **Figure 6b** shows the FEA simulation results of these two models compared with three simulations of the single homogeneous component layer of the A/NC, NG layers and the UFG core. The predications of these two multilayered models show similar load– depth behavior and both fell in between the simulation of the A/NC and the NG layers.

Based on the FEA simulations, the mechanical behavior of the MHS material was explored by deducing each load–depth performance to an effective O–P modulus, effective microhardness and energy dissipation. The effective modulus and hardness predicted for the two models (discrete and gradient systems) showed a loading-dependency, which is not the case for the single homogeneous systems (**Figure 6c** and **d**). The effective modulus and hardness for discrete and gradient systems decreased nonlinearly between that of the A/NC layer at small loads and the values corresponding to UFG core at maximum load of 10 N. Good agreement in the magnitude and the load dependency was achieved between the effective hardness and the experimentally measured microhardness (**Figure 6d**). The effective energy dissipation of these two models was found to increase with increasing load and all fell in between the A/NC layer and the UFG core (**Figure 6e**).

modified the stress distribution and reduced the overall strain values, suppressing crack formation [28]. In our present study, the multilayered models (discrete and gradient) showed a considerable redistribution of the overall equivalent plastic strain field and a significant reduction in the maximum strain levels (**Figure 7**). The plastic equivalent strain contours revealed an increased depth and strain area of plastic deformation for the multilayered systems compared with the monolithic NG material. This is a direct result of transferring the plastic strain to the underlying UFG core with a lower *σY* than the NG layer, thereby diffusing the total plastic deformation energy. The FEA results coincide with the experimental results presented in **Figure 4** where the MHSed Ti diffuses plastic deformation over a greater region relative to the monolithic NG Ti.

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The discrete and gradient models were shown to achieve similar macroscopic effective indentation modulus and microhardness (**Figure 6**). However, FEA simulations revealed differences in the stress and plastic equivalent strain distributions between gradient and discrete models after unloading (**Figure 7**). The magnitude of the equivalent plastic deformation in the top A/NC and NG layers in the gradient system is lower than that experienced in the discrete system. Further, the magnitude and area of plastic deformation in the UFG core are greater in the gradient system compared with the discrete system. This result reveals that the reduction in the stresses and plastic deformation in the top layers is a direct consequence of the increased deformation and energy dissipation accommodated by the softer inner UFG core. Moreover, the mechanical gradations in the successive layers and junctions are observed to give rise to more gradual stress redistribution between component layers, as opposed to the abrupt stress changes observed in the discrete model (**Figure 8**). Such graded stress distribution is believed to lessen the interface failure and increase the interfacial toughness, thus

The different transition patterns between elastic and plastic deformation shown in **Figure 6** are consistent with the FEA simulation results. The smooth transitional region in the MHSed Ti is attributed to its graded stress distribution achieved by microstructural grading. The absence of any interfacial failure, such as delamination or fracture, observed between the layers in the

**Figure 8.** Distribution of Von Mises stress under the indentation regions in (a) discrete and (b) gradient models, while (c)

**4.2. Mechanical grading**

providing strong resistance to loading-damage [10, 29].

shows the Von Mises stresses across the interfaces along the white dotted lines.

#### **3.5. Strain and stress simulation**

The contact load-bearing behavior of the MHSed Ti was further assessed using FEA to simulate the stress and strain distributions within the materials. **Figure 7** shows the stresses and corresponding equivalent plastic strain contours after unloading (maximum load of 1000 N) for the simulated multilayered systems (discrete and gradient) as compared with that of the monolithic NG material. As presented by the color contours, both the stresses and the equivalent strains exhibit graded distributions within the gradient model, as opposed to the abrupt changes observed in the discrete model. The maximum magnitude of equivalent plastic strain in the monolithic NG material (0.17) is greater than those in both the discrete and gradient multilayers (0.14 and 0.13, respectively). However, the multilayered models (discrete and gradient) achieve deeper and a broader plastic deformation field than that of the monolithic NG material. These simulations further suggest that the structural multilayering and grading modified the stress and strain distribution and reduce the overall plastic strain level throughout the material under indentation conditions.
