## 2 Inverse Calculation of Material Constant  
In this section, the material constants are inversely calculated with the steel ball drop test, according to the bending and torsion modes of Bernoulli-Euler beams [34].  
In the bending mode, when a cantilever beam is bent by shear force at the free end, its governing equation is  
\[\frac{\partial^{4}y}{\partial x^{4}}+\frac{1}{a^{2}}\frac{\partial^{2}y}{\partial t ^{2}}=0,a^{2}=\frac{EI}{\rho A} \tag{1}\]  
where \(y\) is the deflection, \(E\) is the Young's modulus, \(I\) is the moment of inertia, \(\rho\) is the density, and \(A\) is the cross-sectional area.  
Separation of variables gives  
\[y(x,t)=Y(x)T(t) \tag{2}\]  
Substituting Equation (2) into Equation (1) leads to  
\[\frac{a^{2}}{Y(x)}\frac{d^{4}Y(x)}{dx^{4}}=-\frac{1}{T(t)}\frac{d^{2}T(t)}{dt^{ 2}}=\omega^{2} \tag{3}\]  
which can be rewritten as  
\[\frac{d^{4}Y(x)}{dx^{4}}-\beta^{4}Y(x)=0 \tag{4}\]  
where  
\[\beta^{4}=\frac{\omega^{2}}{a^{2}}=\frac{\rho A\omega^{2}}{EI} \tag{5}\]  
To solve Equation (4), the following cantilever beam boundary conditions  
\[Y(0)=Y^{\prime}(0)=Y^{\prime\prime}(\updownarrow)=Y^{\prime\prime\prime}( \updownarrow)=0 \tag{6}\]  
are substituted to obtain the characteristic equation:  
\[\cos\beta\updownarrow\cosh\beta\updownarrow=-1 \tag{7}\]  
In the torsion mode, when a cantilever beam is subjected to torsion at the free end, its governing equation is  
\[\frac{\partial^{2}\theta}{\partial x^{2}}=\frac{1}{c_{S}^{2}}\frac{\partial^{ 2}\theta}{\partial t^{2}},c_{S}=\sqrt{\frac{C_{T}}{\rho I}}=\frac{2h}{b}\sqrt {\frac{G}{\rho}} \tag{8}\]  
where \(\theta\) is the torsion angle, \(J\) is the polar moment of inertia, \(C_{T}\) is the torque stiffness, \(b\) and \(h\) are the width and height of the cross section, respectively, and \(G\) is the shear elastic coefficient.  
Let the diameter be  
\[\theta(x,t)=\Theta(x)e^{iwt} \tag{9}\]  
Substituting Equation (9) into Equation (8) we obtain  
\[\Theta^{\prime\prime}+\frac{w^{2}}{c_{S}^{2}}\Theta=0 \tag{10}\]  
To solve Equation (10), the following torsional boundary conditions  
\[\theta(  
In the conducted steel ball drop test, one end of the test piece was clamped and fixed with a jig, the other end was suspended in the form of a cantilever beam, and then the steel ball hit points A and B of the cantilever beam, as shown in Figure 1. A photo of the test setup is shown in Figure 2. The vibration generated by the cantilever beam was measured by a laser Doppler vibrometer (LDV) to obtain time domain velocity-time plots. The obtained data were processed with the fast Fourier transform (FFT) to obtain the frequency domain plots, from which the natural resonance frequency of the test piece could be found. The resonance frequencies corresponding to the first-order bending and torsion modes were then substituted into Equations (5) and (12), respectively, to finally obtain the Young's modulus \(E\) and shear modulus \(G\) of the test piece. Using the following relationship between the Young's modulus and shear modulus,  
\[G=\frac{E}{2(1+\
u)} \tag{13}\]  
the Poisson's ratio \(\
u\) can be obtained. Although Equation (13) is only applicable to isotropic materials, and we only considered orthotropic materials in this study, the material constants were measured only from the length dimension; therefore, Equation (13) can still be used to calculate the value of \(\
u\), which is only the input to the simulation based on finite element method; it was not used in theoretical calculations.  
Figure 1: Schematic of impact points on cantilever beam.