# Nonlinear Impact Force Reduction of Layered Polymers with the Damage-Trap Interface
## 4 Simplified Dynamic Contact Mechanics Analysis  
The low-speed impact problem could be treated as a contact/indentation mechanics problem [22]. For example, [PERSON] and [PERSON] [23] employed [PERSON]'s indentation model to analyze the low-speed impact mechanisms of composite laminates. The impact or indentation load \(P\) of a spherical indenter or projectile with a radius \(R\) is a function of the indentation depth \(h\):  
\[P=\frac{4}{3}\sqrt{R}E_{r}h^{\frac{3}{2}}=C\,h^{\frac{3}{2}} \tag{1}\]  
where \(C\) is the contact stiffness, and \(E_{r}\) is the reduced modulus [24]. [PERSON] et al. [25] found that the maximum impact force is achieved at the zero relative speed of the projectile respect to the target (no penetration), and is simply determined by impact energy \(W\) of the projectile, and the contact stiffness:  
\[P_{max}=1.73\sqrt[5]{W^{3}C^{2}} \tag{2}\]  
Therefore, at a fixed impact energy, decreasing the contact stiffness (proportional to \(E_{r}\)) will decrease the maximum impact force acting on the target (also projectile). The reduced modulus \(E_{r}\) is determined by the [PERSON]'s modulus \(E\) and the [PERSON]'s ratio \(\
u\) of the indenter/projectile material (subscript \(i\)) and the target (subscript \(t\)):  
\[\frac{1}{E_{r}}=\frac{1-\
u_{i}^{2}}{E_{i}}+\frac{1-\
u_{t}^{2}}{E_{t}} \tag{3}\]  
Because decreasing the contact stiffness is equivalent to decreasing the reduced modulus as shown in Equation (1), the through-thickness Young's modulus along the impact direction of the target \(E_{t}\) should decrease according to Equation (3), especially \(E_{i}\) (for steel projectile, 200 GPa) >> \(E_{t}\) (for PMMA, 3.79 GPa). We find that the only factor to reduce \(E_{t}\) is the addition of the soft 5083 adhesive layer. After the soft silicon layers are embedded inside polymers,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Impact Energy (J)**} & \multicolumn{2}{c}{**Max. Impact Force (kN)**} & \multirow{2}{*}{**Percentage Reduction**} \\ \cline{2-2} \cline{4-4}  & **Baseline** & & **Layered Polycarbonate** \\ \hline
7.00 & 7.03 \(\pm\) 0.68 & 3.75 \(\pm\) 1.12 & 46.66 \\
10.00 & 8.16 \(\pm\) 1.26 & 4.50 \(\pm\) 0.95 & 44.85 \\
20.00 & 10.60 \(\pm\) 1.74 & 6.19 \(\pm\) 1.08 & 41.60 \\
30.00 & 12.17 \  
a series rule-of-mixture model [26] could be used to predict the effective through-thickness modulus \(E_{t}\) in terms of polymer volume percent \(V_{p}\) and its Young's modulus \(E_{p}\), and the Young's modulus of the added silicon layer \(E_{add}\) as shown in Equation (4):  
\[\frac{1}{E_{t}}=\ \frac{V_{p}}{E_{p}}+\frac{1-V_{p}}{E_{add}} \tag{4}\]  
The reason to employ a series model rather than a parallel model is that impact load acts on the polymer part first, and then transfers to the adhesive layer, i.e., the same applied load assumption for a series model in mechanics of composite materials ([26]). Since \(V_{p}>90\%\) and \(E_{p}>E_{add}\) (less than 0.001 GPa of the silicon layer based on the adhesive company's data), or \(V_{p}/E_{p}\) is very small, \(E_{t}\approx E_{add}/(1-V_{p})\) in Equation (4). Because the added adhesive layer has a very low Young's modulus, the reduced modulus and the contact stiffness decre. Therefore, the low Young's modulus of the adhesive layer is identified as a major material parameter in order to reduce the maximum impact force and damage.