Thermo-Mechanical Modeling and Simulation of Impact and Solidification of an Aluminum Particle

The heat transfer equations are integrated using the explicit forward-difference time integration rule in ABAQUS/Explicit as follows [23]:

$\theta_{(i+1)}^N=\theta_{(i)}^N+\Delta t_{i+1} \dot{\theta}_{(i)}^N$     (1)

where, $\theta^N$ denotes the temperature at node N and i denotes the increment in a certain dynamic step. At the start of the increment, the values of $\dot{\theta}_{(i)}^N$ are determined by:

$\dot{\theta}_{(i)}^N=\left(C^{N J}\right)^{-1}\left(P_{(i)}^J-F_i^J\right)$     (2)

where, P^J denotes the applied nodal source vector, F^J is the internal flux vector, and C^NJ denotes the lumped capacitance matrix.

$[c]=\int_V[N]^T \rho\left(\frac{d H}{d T}\right)^t[N] d V$     (3)

According to the research [24], the specific heat c_p in the zone of solidification is as follows:

$c_p(T)=\frac{d H}{d T}-\frac{H_f}{\left(T_{l i q}-T_{s o l}\right)}$     (4)

The liquidus temperature, solidus temperature, and latent heat of fusion are represented by the letters T_liq, T_sol, and H_f, respectively. The equations of motion of the body in ABAQUS/Explicit are integrated using the explicit central-difference integration rule [23].

$\dot{\mu}_{(i+1)}^N=\dot{\mu}_{\left(i-\frac{1}{2}\right)}^N+\frac{\Delta t_{i+1}-\Delta t_i}{2} \ddot{\mu}_{(i)}^N$     (5)

$\mu_{(i+1)}^N=\mu_{(i)}^N+\Delta t_{i+1} \dot{\mu}_{\left(i+\frac{1}{2}\right)}^N$     (6)

where, $\mu^N$ denotes a degree of freedom (a component of displacement or rotation), and i denotes the increment number in an explicit dynamic step. In the sense that the kinematic state is advanced using known values of $\dot{\mu}_{\left(i-\frac{1}{2}\right)}^N$ and $\ddot{\mu}_{(i)}^N$ from the preceding increment, the central-difference integration operator is explicit.

According to Hooke's law [25], the elastic strain increment vector $\left\{\Delta \varepsilon^{e l}\right\}$ and the stress increment $\{\Delta \sigma\}$:

$\{\Delta \sigma\}=[D]\left\{\Delta \varepsilon^{e l}\right\}$     (7)

where, [D] holds the elastic constants for the Poisson's ratio (ν) and the temperature-dependent elastic modulus (E):

The total strain vector $\{\Delta \varepsilon\}$ can be represented using the following formula.

$\{\Delta \varepsilon\}=\left\{\Delta \varepsilon^{e l}\right\}+\left\{\Delta \varepsilon^{p l}\right\}+\left\{\Delta \varepsilon^{t h}\right\}$     (8)

The elastic strain increment vector is represented by $\left\{\Delta \varepsilon^{e l}\right\}$, the plastic strain increment vector by $\left\{\Delta \varepsilon^{p l}\right\}$, and the thermal strain increment vector by $\left\{\Delta \varepsilon^{\text {th }}\right\}$.

This formula can be used to get the thermal strain [26]:

$\left\{\Delta \varepsilon^{t h}\right\}=[B] \Delta T$     (9)

where, [B] represents the strain function matrix and [T] represents the temperature difference between the current and reference temperatures.

The following expression is used in this investigation to compute the heat produced by plastic work [27]:

$\int_{T_0}^T \rho C_P \mathrm{~d} T=\beta \int_{\varepsilon_{P 0}}^{\varepsilon_P} \sigma \mathrm{d} \varepsilon$     (10)

where, β is the inelastic heat fraction, denoting a heat flux per unit volume, and the default value is 0.9 in ABAQUS/Explicit. The subscript $0$ denotes an initial value.

The current numerical model simulates the impact and deposition or flattening of a single aluminum particle on steel substrate during thermal spray process using a dynamic-temperature-displacement-explicit approach. We start off the first section of this study by validating the work that has already been done. The same issue was addressed using the volume of fluid method as well, Xue et al. [8]. Figures 2 and 3 demonstrate that the behavior of the present results, particularly during the first moments of contact, behave similarly to those of Xue et al. [8]. The two investigations used different approaches, which explains the discrepancy in the form and spread factor of the flattened particle.

At an initial temperature of 200℃, the results show the temperature progression of a 3.92 mm-diameter aluminum particle impacting on a tool steel substrate. Initial values of velocity and temperature of the particle are 3 m/s and 630℃, respectively. When the particle makes contact with the substrate, it spreads, solidifies, and takes on a lamellar shape. 

Table 2. Mechanical properties of aluminum 6111

Table 3. Thermal properties of aluminum 6111

Table 4. Latent heat, solidus and liquidus temperature of aluminum 6111

Density of AL 6111 is 2700 m3/kg.

Table 5. Mechanical properties of steel

Table 6. Thermal properties of steel 6111

According to the results, heat transfer from the particle to the substrate starts as soon as there is contact, and as a result, the temperature of the substrate beneath the splat surface increases from 200°C to 516°C in 0.6 ms before progressively decreasing; for more details, see Figure 4. Additionally, it is clear from Figure 4 that the maximum error between the experiment and simulation of Xue et al. [8] is about 15.62%. This difference is due to the assumption of using a constant melting point during simulation and the presence of contaminants and an oxide layer in experiment.

Figure 5 shows the contours of total displacement for various times. The particle starts to spread laterally as it makes contact with the substrate, creating a lamellar structure. It can be seen that the impact zone has a small displacement. The highest displacement always happens at the splat's edge. It is also evident that the particle's spreading begins to progressively slow down around about 2ms. The dissipation of the particle's kinetic energy through elastic and plastic deformation work results in a slowdown of the particle's velocity.

The Von-Mises stress over time is shown in Figure 6 to explain the mechanical interaction between the splat and the substrate. When a particle is in touch with the top face of the substrate, where the stress is initially stronger, the stress gradually spreads through the substrate over time. 

Furthermore, it is clear from this figure that the substrate is subjected to higher stress than the particle. These findings can be explained by the different young modulus values for the two materials, as reported by Benramoul and El-Hadj [20]. 

A thin layer close to the surface experiences frictional shear as a result of particle spreading at the same level as the surrounding zone. There is residual tensile stress close to the surface as a result of the interplay of these layers. The XY shear stress contours are depicted in Figure 7. The findings demonstrate that shear stress grows over time with symmetry in intensity, with the maximum shear stress always occurring near the substrate's edge.

figure_2a.png

(a) t=0.1 ms

figure_2b.png

(b) t=0.3 ms

figure_2c.png

(c) t=0.7 ms

figure_2d.png

(d) t=2.0 ms

figure_2e.png

(e) t=3.0 ms

figure_2f.png

(f) t=7.0 ms

Figure 2. Comparison of the numerical model used with other numerical and experimental models to validate the sequential effect of a 3.92mm aluminum drop

figure_3.png

Figure 3. Comparison of the spread factor between the current simulation's results and those reported in the literature for an aluminum droplet impacting at 3m/s

figure_4.png

Figure 4. Substrate surface temperature histories under the splat center

figure_5a.png

(a) t=0.1 ms

figure_5b.png

(b) t=0.3 ms

figure_5c.png

(c) t=0.7 ms

figure_5d.png

(d) t=2.0 ms

figure_5e.png

(e) t=3.0 ms

figure_5f.png

(f) t=7.0 ms

Figure 5. Contour plots for displacement following successive impacts of a 3.92mm-diameter aluminum droplet impacting at a 3m/s impact velocity

figure_6a.png

(a) t=0.1 ms

figure_6b.png

(b) t=0.3 ms

figure_6c.png

(c) t=0.7 ms

figure_6d.png

(d) t=2.0 ms

figure_6e.png

(e) t=3.0 ms

figure_6f.png

(f) t=7.0 ms

Figure 6. Contour plots for Von mises stress following successive impacts of a 3.92mm-diameter aluminum droplet impacting at a 3m/s impact velocity

figure_7a.png

(a) t=0.1 ms

figure_7b.png

(b) t=0.3 ms

figure_7c.png

(c) t=0.7 ms

figure_7d.png

(d) t=2.0 ms

figure_7e.png

(e) t=3.0 ms

figure_7f.png

(f) t=7.0 ms

Figure 7. Contour plots for XY shear stress following successive impacts of a 3.92mm-diameter aluminum droplet impacting at a 3m/s impact velocity

When a material is subjected to a load exceeding a threshold called the yield strength, it undergoes permanent and non-reversible deformation. The transition from elastic behavior to plastic behavior is known as yielding. Figure 8 displays the contours of the effective plastic strain (PEEQ). The plastic strain has larger values in the particle section than stress does. The impact zone of the particle is where the plasticity first manifests itself. Then, as the particle flattens, the plasticity region expands to encompass the entire particle. Additionally, it was discovered that the effective plastic strain had maximum values at both the center and the edges of the splat.

figure_8a.png

(a) t=0.1 ms

figure_8b.png

(b) t=0.3 ms

figure_8c.png

(c) t=0.7 ms

figure_8d.png

(d) t=2.0 ms

figure_8e.png

(e) t=3.0 ms

figure_8f.png

(f) t=7.0 ms

Figure 8. Contour plots for PEEQ following successive impacts of a 3.92mm-diameter aluminum droplet impacting at a 3m/s impact velocity

Figure 9 showed the change in contact pressure over time at the particle's impact zone's center. In the first moments of contact, the contact pressure between the particle and the substrate rapidly increases and reaches a high of about 40 KPa. This low value is the result of the particle’s plastic deformation absorbing energy. Furthermore, contact pressure varies between positive and negative values, the opposite of the conclusions of the VOF model [8]. The splat's adherence to the substrate is encouraged by the positive contact pressure. However, the negative value has a rebound impact on the particle, according to Benramoul and El-Hadj [20]. Furthermore, the negative contact pressure at 0.002 s is most likely due to particle contraction during solidification or possibly the reaction of the substrate after getting impacted by the particle.

figure_9.png

Figure 9. The change in contact pressure at the particle's center contact point