# Sintering of Two Viscoelastic Particles: A Computational Approach
## 6 Results  
### Effect of the Initial Geometry  
From the governing equations of the Newtonian constitutive behavior follows that the initial geometry and configuration are the only factors that affect the flow. To analyze the influence of different initial contact radii, we define four geometries: \(R=1\) and \(y_{\mathrm{n}}=[0.125,\,0.25,\,0.4,\,0.5]\). Without loss of generality, we set the viscosity \(\mu=1\) and the surface tension \(\gamma=1\). The results are shown in Figure 5, where the contact radius \(y\) is depicted in time \(t\). Subsequently, we shift the graphs of \(y_{\mathrm{n}}=0.25\), \(y_{\mathrm{n}}=0.4\) and \(y_{\mathrm{n}}=0.5\) in time such that the initial contact radii coincide with the curve of \(y_{\mathrm{n}}=0.125\). The results are given in Figure 6, in which we see that all lines overlap. This indicates that the shape evolution of the contact radius \(y\) is independent of the flow history, if the round off parameter \(R_{\mathrm{n}}=R\times(y_{\mathrm{n}}/R)^{3}\)[20] is used. Figure 5: The evolution of the contact radius \(y\) in time \(t\) of two equal spheres with different initial contact radii \(y_{\mathrm{n}}=[0.125,\,0.25,\,0.4,\,0.5]\), using the Newtonian constitutive equation. Figure 6: The evolution of the contact radius \(y\) in time \(t\) of two equal spheres with different initial contact radii \(y_{\mathrm{n}}=[0.125,\,0.25,\,0.4,\,0.5]\), using the Newtonian constitutive equation. The lines of \(y_{\mathrm{n}}=0.25\), \(y_{\mathrm{n}}=0.4\) and \(y_{\mathrm{n}}=0.5\) are shifted in time such that the initial contact radii coincide with the graph of \(y_{\mathrm{n}}=0.125\). Next, we keep the initial contact radius constant \(y_{\rm n}=0.4\), and we change the round off parameter \(R_{\rm n}=[R\times(y_{\rm n}/R)^{3},\ (R/2)\times(y_{\rm n}/R)^{3},\ (R/4)\times(y_{\rm n}/R)^{3}]\). The results are given in Figure 7, where the contact radius \(y\) is depicted in time \(t\). Since all curves coincide, we can conclude that the influence of the round off parameter \(R_{\rm n}\) on the shape evolution of the contact radius \(y\) is negligible. Furthermore, the curvature \(\kappa\) at point \((0,\,y)\) at the surface of the neck is shown in time \(t\) for different round off parameters \(R_{\rm n}=[R\times(y_{\rm n}/R)^{3},\ (R/2)\times(y_{\rm n}/R)^{3},\ (R/4)\times(y_{\rm n}/R)^{3}]\) in Figure 8. Following [PERSON] and [PERSON] (2010), the curvature is calculated from the contour curve \(r=g(z)\) of surface \(\Gamma\), using:  
\[\kappa=\frac{1}{R_{1}}+\frac{1}{R_{2}}, \tag{45}\]  
with \(R_{1}\) and \(R_{2}\) the two principal radii of curvature:  
\[R_{1} =\frac{(1+g^{\prime 2})^{3/2}}{g^{\prime\prime}}, \tag{46}\] \[R_{2} =-g\sqrt{1+g^{\prime 2}}. \tag{47}\]  
Herein, \(g^{\prime}=dg/dz\) and \(g^{\prime\prime}=d^{2}g/dz^{2}\). From the curvature, the Laplace pressure \(p_{\rm L}\) can be calculated using \(p_{\rm L}=\gamma\kappa\), which is equal to the radial component of the Cauchy stress tensor \(\sigma_{rr}\) at the surface. From Figure 8, it follows that the curvature \(\kappa\) at point \((0,\,y)\) increases with respect to the initial geometry until it reaches a maximum value and subsequently decreases. This holds for all different values of the round off parameter \(R_{\rm n}=[R\times(y_{\rm n}/R)^{3},\ (R/2)\times(y_{\rm n}/R)^{3},\ (R/4)\times(y_{\rm n}/R)^{3}]\). This behavior is depicted in Figure 9, where the contour line \(r=g(z)\) of the two particles is shown at times \(t=[0,\,0.08,\,0.2]\), using \(R_{n}=R\times(y_{\rm n}/R)^{3}\). Herein, \(t=0.08\) is the time at which the curvature \(\kappa\) reaches its maximum value. From this, we can conclude that the choice of the round off parameter \(R_{\rm n}\) strongly influences the evolution of curvature and, therefore, the evolution of the local stresses in the material.