# Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams
## 3 Closed-Form Solution of [PERSON]'s Two-Phase Integral Model Equations
### Clamped Pinned Beam (CP)  
In this section, we look at a clamped pinned beam in which case the boundary conditions are  
\[w(0)=w^{\prime}(0)=0,\quad w(L)=M(L)=0. \tag{34}\]  
To solve analytically the integro-differential boundary value problem (16), (34), we define the operator \(B:X\to X\) as  
\[Bw(x) = \frac{d^{2}}{dx^{2}}\biggl{(}\frac{d^{2}w(x)}{dx^{2}}+\frac{ \xi_{2}}{\xi_{1}}\int_{0}^{L}k(x,t)\frac{d^{2}w(t)}{dt^{2}}dt\biggr{)},\] \[\mathcal{D}(B) = \Bigl{\{}w(x)\in C^{4}[0,L]:w(0)=w^{\prime}(0)=0,\;w(L)=0,\] \[\biggl{[}\frac{d^{2}w(x)}{dx^{2}}+\frac{\xi_{2}}{\xi_{1}}\int_{0} ^{x}k(x,t)\frac{d^{2}w(t)}{dt^{2}}dt\biggr{]}_{x=L}=0\biggr{\}},\]  
where the definition (14) is utilized, and write (16), (34) in the symbolic form  
\[Bw(x)=\frac{1}{EI\xi_{1}}q(x),\quad 0<x<L,\quad q(x)\in X. \tag{35}\]  
We define the Fredholm integro-differential operator \(B_{2}:X\to X\) as in (22) in Section 3.2, namely  
\[B_{2}w(x) = \frac{d^{2}w(x)}{dx^{2}}+\frac{\xi_{2}}{\xi_{1}}\int_{0}^{L}k(x,t )\frac{d^{2}w(t)}{dt^{2}}dt,\] \[\mathcal{D}(B_{2}) = \Bigl{\{}w(x)\in C^{2}[0,L]:\;w(0)=w^{\prime}(0)=0\Bigr{\}}, \tag{36}\]  
and the differential operator \(B_{1}:X\to X\) as  
\[B_{1}y(x) = \frac{d^{2}y(x)}{dx^{2}},\] \[\mathcal{D}(B_{1}) = \Bigl{\{}y(x)\in C^{2}[0,L]:y(L)=0\Bigr{\}},\]  
where \(y(x)=B_{2}w(x)\).  
Then, the operator \(B_{1}B_{2}:X\to X\) is defined on  
\[\mathcal{D}(B_{1}B_{2}) = \Big{\{}w(x)\in C^{2}[0,L]:w(0)=w^{\prime}(0)=0,\,y(x)=B_{2}w(x)\in \mathcal{D}(B_{1})\Big{\}} \tag{37}\] \[= \Big{\{}w(x)\in C^{2}[0,L]:w(0)=w^{\prime}(0)=0,\,y(x)\in C^{2}[0, L],\,y(L)=0\Big{\}}\] \[= \Big{\{}w(x)\in C^{4}[0,L]:w(0)=w^{\prime}(0)=0,\] \[\quad\left[\frac{d^{2}w(x)}{dx^{2}}+\frac{\xi_{2}}{\xi_{1}}\int_ {0}^{x}k(x,t)\frac{d^{2}w(t)}{dt^{2}}dt\right]_{x=L}=0\Big{\}},\]  
and the boundary value problem (35) becomes  
\[Bw(x) = B_{1}B_{2}w(x)=\frac{1}{EI\xi_{1}}q(x),\quad 0<x<L,\quad q(x) \in X,\] \[\mathcal{D}(B) = \{w(x)\in\mathcal{D}(B_{1}B_{2}):w(L)=0\}. \tag{38}\]  
The solution of the problem  
\[\text{DBVP:}\quad B_{1}y(x)=\frac{1}{EI\xi_{1}}q(x),\quad 0<x<L,\quad q(x) \in X,\]  
for any \(q(x)\in X\) is given by  
\[y(x)=\frac{1}{EI\xi_{1}}\bigg{[}\int_{0}^{x}(x-t)q(t)dt-\int_{0}^{L}(L-t)q(t) dt+C_{1}(x-L)\bigg{]}, \tag{39}\]  
where \(C_{1}\) represents an arbitrary constant.  
By using \(y(x)\) in (39), we solve the problem  
\[\text{FIDBVP:}\quad B_{2}w(x)=y(x),\quad 0<x<L,\]  
which in the case of a kernel function \(k(x,t)\) of the type (15) degenerates to the problem  
\[\text{VFIDBVP:}\quad B_{2}w(x)=y(x),\quad 0<x<L, \tag{40}\]  
where operator \(B_{2}\) is given in (27). Working just like in Section 3.2 except that now \(f(x)=y(x)\) as in (39), we get the solution  
\[w(x)=D^{-1}f(x)+D^{-1}g(x)W^{-1}\Psi\Big{(}AD^{-1}f(x)\Big{)}, \tag{41}\]  
which depends linearly on the arbitrary constant \(C_{1}\).  
By requiring \(w(L)=0\), we can calculate \(C_{1}\) which when replaced at (41) gives the exact solution of the boundary value problem (38) or the nonlocal Euler-Bernulli Equation (16) subject to the boundary conditions (34).